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 × interval operation is applied. This rapid extension is not acceptable in many applications. To overcome this problem, the author of this chapter presented another set of arithmetic operations on fuzzy numbers (Alex, 2007). Since the new arithmetic operations on fuzzy numbers are different from the existing operations, the fuzzy supply chain analysis based on the new set of arithmetic operations is different from the fuzzy supply chain analysis introduced earlier. That is why the author has presented his modeling of fuzzy supply chains based on the earlier work here as a supplement to works on the fuzzy supply chains.

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) ci provides/produces a certain kind of part/product pj at a certain unit/factory. For simplicity, assume that different units provide different kinds of parts/products. Let C={c1,c2,⋯,cn} be the set of all sites in a supply chain, and C* be an extension of such that it includes the set of external suppliers denoted by Y and the set of end-customer centers denoted by Z :

We will simply treat an external supplier or an end-customer center also as a site. There is a relationship among the sites of C* : If a site ci uses materials/parts/products from a site cj , then we say the site cj supplies the site ci and is denoted as cj→ci . The site cj is called an up-site of ci , and ci is called a down-site of ci . The suppliers in Y have no up-sites and the customers in Z have no down-sites in C* . The relation of supplying can be described in mathematics as a subset S⊆C*×C*

If we do not consider the case of a site supplying itself, then the supplying relation S is anti-reflexive, i.e., for any cj∈C* cj→cj is not possible. If we do not consider the case of two sites supplying each other, then S is anti-symmetric, i.e., for any ci cj∈C* , if ci→cj , then cj→ci is not possible.

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 Sn will become an empty set when n is large enough. Let h be a number large enough such that Sh is empty. Set

For any site cj∈C , let Dj and Uj be the set of down-sites and up-sites of cj , respectively. Suppose that Dj1=Dj . For any n1 , set

The sites belonging to Djn and Ujn are called the n-generation down-sites and up-sites of cj , respectively. Clearly, any down-site is the 1-generation down-site, and any up-site is the 1-generation up-site. It is obvious that Djn or Ujn may become an empty set when n is large enough. Set

These are the enclosures of Dj and Uj , and are called the down-stream and up-stream of cj , respectively.

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 c1 be a site supplying sugar, c2 be a site supplying the cake mix for cakes, and c3 be the site supplying the birthday-cakes. We have that c1→c2 c2→c3 , and c1→c3 _. Since c1 is the up-site of c2 and c2 is the up-site of c3 , so that c1 is the 2-generation up-site of c3 . But c1 is also the first generation up-site of c3 . So that U31∩U32≠φ . Such situations may bring complexity to the research.

For a simple supply chain (C*,S) , any site can be in at most one generation of upstream and at most one generation of downstream of another site.

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 cb∈B Ub should contain at least an external supplier:

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

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

It is obvious that all sites in C can be divided as different up-generations of c0 . If cj∈U0n , we say that the (generation) code of cj for c0 is n, and denoted as χ=jχj0=n . Since the supply chain is simple so that for any site cj in C with code n, there is one and only one linear chain connecting the site cj and c0 given by:

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 cj in C has a code χj0 for a root-site c0 if cj→c0 , and has one and only one linear chain connecting cj and c0 .Case 2 is a generalization of case 1, and the case 3 is a generalization of case 2. In the rest of the chapter, we will limit our attention to case 2 of a simple supply chain.

For each site cj in C, let qji(t) be the order quantity of pj -part/material from the down-site ci , which is called the order-away quantity of cj at time t. While qkj(t) , the pk -part/material quantity in up-site ck ordered by cj , is called the order-in quantity of cj at time t.

The following review period policy is assumed here: For any site cj in C, the time of ordering in the up-parts could not be arbitrary, but limited at tj tj+Tj tj+2Tj,⋯ . These timings are called the review times, and Tj0 is called the review period of cj . To be simple, assume that tj=0 for any cj in C.

For any site cj∈C , suppose that cj→ci . Set

is the number of pj -parts that has been ordered to be sent out to the down-site of cj during the last period (n−1)Tj≤tnTj and is called the passed away number of pj ’s in the last period. Set

This is called the order-away rate of pj at the time t. For a root-site c0 , the passed-away number of p0 –products is called the demand number at time t denoted as d(nT0)=α0(nT0) Set

This is called the demand rate of p0 at the time t.

Suppose that each pi -product/part is produced by means of wji pieces of pj -parts, we call wji the equivalence of a pi -part for the pj -part. For any site pair (cj,ci)∈S , there is an equivalence value wji , which reflects the production ingredient of down-products by means of the up-parts.

In case 2, for any site cj in C with code χ=jn , there is one and only one linear chain connecting it to its root site c0 as:

This is called the equivalence of a product for the pj -part. The production of each final product p0 needs wj pieces of pj -parts to supply it.

The main problem in supply chain analysis is: How to set up the reasonable inventory levels in all sites of C ? Let Ij=Ij(t) be the real inventory of pj -parts of site cj at time t=nTj . This should be a negative number whenever it is in shortage at the time. We do not want a site to be in the shortage, so we want that Ij >0; While its value should not be too high since then there will be a high inventory maintenance cost; The goal of supply chain management is to minimize the supply chain inventory cost and to limit the possibility of shortage as much as possible.

The expected inventory level of the site cj at the time t=nTj should be responsible not only for supplying the down-site of cj during the next period [nTj,(n+1)Tj] , but also for a longer time until the birth of the next batch of pj -parts produced from up-parts ordered in cj at the next review time t'=(n+1)Tj . The length from t=nTj to the mentioned time can be denoted as

This is called the looking time of cj ; while Lj is called the replenishment time of cj . The concrete expression of Lj is

where

which stands for the reasonable inventory level of site cj at time t=nTj Sj is called the order-up-to level of site cj at time t=nTj

which is the real order of pk -parts from site cj at time

The main task in supply chain analysis is the determination of the order-up-to levels { Sj } (j=1,...,n) in all sites of the chain at a time t.

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 parameter α

Let a real number a be the answer, then we say that the fuzzy parameter α has the estimation value a , denoted as a=m(α)

Question 2: What is the confidence on your estimation for α Please place the mark on a proper location in the real number line that represents the confidence interval [0, 1]. The expert points out a mark at the proper position in the interval [0, 1] to represent the degree of his confidence on the estimation of the number in question 1. For example, according to the location of the mark shown in the Fig. 1, we can get a real number ϕ =0.75, which is called as the confidence degree of the expert on his estimation.

Figure 1.

The confidence on the parameter estimation

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 ρ∈[0,1] to represent his skill degree on subjective estimation. The closer the score value is to 1 the higher the authority. The score can be measured and adjusted by the success rate in practical situations. ρ is called the authority index of the expert. The product of the authority index ρ of an expert and the confidence degree ϕ of his estimation on a fuzzy parameter represents the subjective accuracy of this estimation, denoted as τ=ρ×ϕ We call δ=1−τ the ambiguity degree of the estimation. A fuzzy parameter α can be represented by a pair of two real numbers, its estimation value a and its ambiguity degree δ

α = a1±δ

0≤δ≤1

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The ambiguity degree of the parameter α could also be called the estimation error of the estimation in α and denoted as δ=e(α) . The formula (3.1) looks like the representation of error in measurement theory. Yes, they are very similar. The only difference is: The error in measurement is caused by the impreciseness of instruments and observation; while the ambiguity is caused by the fuzziness in subjective estimation. In the error theory, there are two kinds of errors: absolute error and relative error. The ambiguity reflects the error in subjective estimation and it is not an absolute error, but a relative error. The relative error plays a more essential role. For examples, when we estimate that the height of the wall as 2±0.2 units, the estimation value is a=2 units and the absolute error is a×δ=0.2 ; when we estimate that the length of the street is 2000±200 units, the estimation value is a=2000 units and the absolute error is a×δ=200 ; when we estimate that the length of an insect is 0.002±0.0002 units, the estimation value is a=0.002 and the absolute error is a×δ=0.0002 .There are differences in the three examples, but the relative error is the same δ=0.1 . The estimation errors are invariable on the changing of unit. It reflects the intrinsic quality of subjective estimation.

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 r=|a|×δ

μα(x)={0if−∞x≤a−r1+x−arifa−rx≤a1−x−arifax≤a+r0ifa+rx∞

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Since 0≤δ≤1 a fuzzy parameter is a special triangle fuzzy number whose radius is r≤|a|

Figure 2.

Example of 10 fuzzy parameters

In the Fig. 2, we can see a set of fuzzy parameters with estimation value a=100 have membership functions shown as the broken lines ATA,BTC,DTE,FTG,and OTH with ambiguity δ=0,0.25,0.5,0.75,and 1 respectively; those fuzzy parameters with estimation value a'=−100 have membership functions shown as the broken lines A'T'A' B'T'C' D'T'E' F'T'G' , and O'T'H' with ambiguity δ=0,0.25,0.5,0.75,and 1 , respectively.

Definition 3.1 Given a positive real number 0*1, we call V, the set of fuzzy parameters α=a±r with r|a|≤δ* , the δ* -systems of fuzzy parameters.

For example, suppose that V is a 0.05-system of fuzzy parameters. The fuzzy parameter 2±1∉V since r|a|=0.50.05 The fuzzy parameter 1±0.05∈V since

r|a|=0.05

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.

Figure 3.

The *-system of fuzzy parameters

In the Fig. 3, the radius of the fuzzy parameter −3±3δ* is 3δ* the radius of the fuzzy parameter −2±2δ* is 2δ* and the radius of the fuzzy parameter −1±δ* is δ* . The radius of fuzzy parameter 1±δ* is δ* ; the radius of fuzzy parameter 2±2δ* is 2δ* and the radius of fuzzy parameter 3±3δ* is 3δ* As we see from figure 3, the estimation values closer to zero, the narrower the membership function width; the estimation value farther away from zero, the wider the membership function width. However, the ambiguities of the fuzzy parameters in a δ* -system are all restricted by *. A δ*− system includes not only those fuzzy parameters whose ambiguities are equal to δ* but all fuzzy parameters whose ambiguities are less than δ* . The δ*− systems are not disjoint but expanded when the parameter δ* is increasing: δ1*− system V1⊆ δ2*− system V2

(δ1≤δ2)

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Proposition 3.1 Suppose that V is a δ* -system of fuzzy parameters, where 0≤δ*≤1 For any non-zero fuzzy parameter α=a±r∈V , the support of α does not contain zero as an inner point. i.e., 0∉(a−r,a+r) Proof Assume that 0∈(a−r,a+r) If a0 , then 1−ra01+ra Then 1−δ*≤1−ra0 , i.e., δ*1 . This is a contradiction to the requirement of δ*≤1 . Suppose that a0 , then 1−ra01+ra Since δ*≥|ra|=−ra 01+ra≥1−δ* , i.e., δ*1 This is a contradiction with the requirement of δ*≤1 . According to the reduction to absurdity, the assumption is not true. So 0∉(a−r,a+r)

Using Proposition 3.1, we can say that a fuzzy parameter α is positive if the estimation value of α is positive, and α is negative if the estimation value of α is negative. Proposition 3.1 constrains the fuzzy parameters in our δ− system in pure sign, i.e., the support of any fuzzy parameter does not contain zero. This is not a real constraint in practical but reflects such a faith in the thinking process: Human beings like to do fuzzy estimation on “how much” but not fuzzy on the main direction to do it. For example, suppose we are telling somebody: “To go to the post office, turn left and go about 150 meters”. It may be acceptable if the distance is not estimated precisely; the distance is not exactly 150 meters, instead it is 164 meters. But it is not acceptable if the direction to turn left is wrong. A δ− system is free in use if we put the zero point in such a place from where the directions toward West and East are distinguished.

It is worth noting that the ambiguity δ of a fuzzy parameter α could be larger than zero whenever its estimation value a=0 In this case, α=a±|0|×δ=a±0 Indeed, for a fuzzy parameter with estimation value zero, it can have arbitrary ambiguity δ

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.

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

We observe that the value of qji(t) , the order-away quantity of cj at time t, is not known yet. If it is not deterministic, then uncertainties occur when we take estimation on this value. As mentioned earlier, there are two kinds of uncertainties: randomness and fuzziness. If there are enough data representing the past values and the conditions related to those data are continuing onto the present, then the value qji(t) could be treated as a random variable; otherwise, if there is not enough statistical data available, or if the conditions have been changed, it could be better treated as a fuzzy parameter and be estimated as mentioned in Section 3. As a supplement to the existing treatments of uncertainty in supply chain analysis, this chapter presents a new phase of the uncertainties treatment when we encounter randomness and fuzziness simultaneously. Accordingly, no matter whether the parameter is a random variable or a fuzzy number, it is always treated as a fuzzy parameter:

Here E(qji(t)) is the mathematical expectation of qji(t) and σ(qji(t)) is the root-mean-square error of qji(t) . Why does the author treat a random variable as a fuzzy parameter? Because the core modeling for fuzzy supply chain analysis is imitating the experts’ experiences. Experts responsible for fuzzy supply chain analysis apply their skill at two stages: 1. Estimating value of each involved parameter; 2. Choosing of arithmetic operations on fuzzy parameters according to the estimation-error-limitation principle. In the first stage, if the estimated value is the mathematical expectation of a random variable, then the expert could rely on the objective methods in probability theory and get the resulting value. That is fine! It could save expert’s time to do subjective estimation. Whenever the fuzzy estimations have been input into the second stage, the root-mean-square error has been transferred into the estimation error in the fuzzy parameters’ operations. This will not involve operations on the probability distributions of random variable. Apart from taking the operations ± on independent variable, there may not be any need to do rigid probabilistic operations on random variables’ in the practical applications.

Similarly, the order-away quantity of cj at time t, aj(t) , is also a fuzzy parameter no matter it is a random variable or a fuzzy number.

The order-away rate of pj at the time t α¯j(t) , is also a fuzzy parameter no matter it is a random variable or a fuzzy number.

where

For a root-site c0 , the demand number at time t is treated as a fuzzy parameter d(t)=m(d(t))(1±e(d(t))) . The demand rate of p0 at the time t, treated as a fuzzy parameter

The material lead time from ck to cj Mkj is also to treated as a fuzzy parameter no matter it is a random variable or a fuzzy number.

The delay time of pk -parts for cj Gkj is also to be treated as a fuzzy parameter:

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 τj , the estimated number of occurrences of downtime ϕj , the duration of downtime on the production line ϑj . While the production capacity Cj is a deterministic number, it could be also treated as a fuzzy parameter provided we take m(Cj)=Cj and e(Cj)=0 . The replenishment time of cj Lj is also treated as a fuzzy parameter

where

The order-up-to level for site cj at time t, Sj(t) is a fuzzy parameter, which is the product of αj(nTj) and

where

We note that shortage may occur whenever

where nTj≤tnTj+Lj(t) . It implies that the shortage interval could be roughly written as ( −∞,Sj ). But since Sj is a fuzzy number, the interval is not a crisp interval. To conveniently control the inventory, we need to pick out two thresholds from Sj : two real numbers Sjo and Sjp called the optimistic and the pessimistic order-up-to levels of site cj , respectively. They are determined by the following equations:

where r is a given fill rate, A typical value is r=0.95 ; and

These are called the left possibility function and the necessary function of fuzzy variable Sj respectively. We have a formula to get the two real order-up-to levels from the fuzzy order-up-to level:

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:

Just as the stationary random process has a stationary distribution, a stationary supply chain has a stationary possibility distribution with a constant demand rate d¯ Of course, the real demand from the customers is still a variable. No mater how complex the supply chain system is, we can think of it as a network of water flow. In the water flow network, we will have a stationary flow whenever the input equals the output at every node. To maintain a stationary flow in a supply chain network, the best way is for any site to know how many units passed away during the last period; how many units should be ordered back in the review time. Here comes the stationary strategy.

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.

Proof. We use the principle of mathematical induction for the code n=χ(cj)

Assume that the Proposition is true for any cj with code n=χ(cj)=1 Indeed, implies that cj→c0 . It is obvious that (5.2) is true for the base case.

Suppose that (5.2) is true for n, we are going to prove that it is true for n+1 Suppose that cj→ci and χ(cj)=n then αj(t)=wi×d¯ We have that

So (5.2) is true.

According to (4.20), we have

Since the chain is in stationary, we can write m(Lj) in detail according to (4.1)-(4.18) and get

According to (4.24), we get