# Operational Complexity of Supplier-Customer Systems Measured by Entropy—Case Studies

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theoretical Background

#### 2.1. Entropy Used to Measure Operational Complexity

_{i}≥ 0, with ${{\displaystyle \sum}}_{i=1}^{N}{p}_{i}=1.$ The Shannon entropy is given by following formula, which corresponds to Equation (1) taking c = 1 and b = 2:

_{2}($x$) in particular, whilst the entropy calculated with natural logarithms log($x$), i.e., with b = e, will be denoted by S from here on.

_{BGS}and called Boltzmann–Gibbs–Shannon entropy, in the most correct sense.

^{2}, which helps significantly to elucidate Equations (7) and (9).

^{®}(version 10.1) to perform our numerical experiments with (c,d)-entropy given by Equation (8), as well as to discover a more general relation between (c,d)-entropy and S

_{BGS}than given by Equations (7) or (9), respectively.

^{2}, where × denotes a Cartesian product of two sets. The distribution $\varphi $ is constructed from finite samples of random numbers generated by integer pseudo-random generators. In general, L will denote the length of list of such numbers, and their range will be {0, 1, …, k

_{max}}, thus N = k

_{max}+ 1.

_{max}, L], which serves exactly that purpose to give a list of L pseudorandom integers ranging from 0 to k

_{max}desired. Sorting the list for given k

_{max}, the frequencies and corresponding empirical distribution $\varphi $ will be calculated as an approximation of a uniform distribution of probabilistic system with N states.

_{max}. Since the results were very similar, we present just the following ones, see Figure 1 and Figure 2. Raw data of the generated probabilistic system are given in Table 1. In the first row, there are particular system states selected, i.e., ten bins denoted 0, …, 9 in sequel, which makes N = 10 as k

_{max}= 9. In the second row, there are the corresponding frequencies of state occurrences. Further, the Table 2 gives the corresponding probabilities p

_{1}, …, p

_{10}, and thus the distribution $\varphi $ is easily obtained from Table 1.

_{1}$\cup$ Ω

_{2}, we are able to localize the curves more distinctly. The mutually disjoint sub-regions Ω

_{1}, Ω

_{2}, are defined using suitable separating line given by two incidence points, e.g., (1,0) and (5,5), in the following way:

_{1}contains just one curve, denoted λ

_{1}, and called the main branch of the intersection, ${S}_{c,d}^{*}\left(\varphi \right)$ $\cap$ ${S}_{\text{BGS}}\left(\varphi \right)$ over (c,d) $\u03f5$ Ω, while sub-region Ω

_{2}contains two other curves, called secondary branches, on the contrary. The point (c,d) = (1,1), which stands in Equations (7) and (9) [31], thus belongs to λ

_{1}evidently. Concluding, the curve λ

_{1}, i.e., the main branch, is given by:

#### 2.2. Information Scheme of Supplier-Customer Systems and Basic Structure of a Problem-Oriented Database

#### 2.3. Definition of Variables

_{1}, ..., P

_{n}} being handled within a supplier-customer system. In general, there are two types of variables relating quantity and time to be considered for a particular product P

_{i}, i = 1, ..., n. All of them are reported at both the supplier and customer side, and thus, they should be reported at the interface, as well. Monitoring such variables provides time series which form the core of information for measuring the operational complexity of supplier-customer system. A list of typical variables monitored for products P

_{i}, i = 1, ..., n, is given in Table 3.

_{(a,b)}Q

_{i}and

_{(a,b)}T

_{i}, i = 1, ..., n, where a stands for a side (s—supplier, i—interface, c—customer) and b denotes a type of production in particular, i.e., scheduled (s), actual (p), forecast (f), order (o), and delivery (d), thus generalizing the scheme presented in [29].

_{(a,b)}Q

_{i}−

_{(u,v)}Q

_{i},

_{(a,b)}T

_{i}–

_{(u,v)}T

_{i}, (a,b) ≠ (u,v), a,u ϵ {s, i, c}, b,v ϵ {s, p, f, o, d}

- Recast such data into a suitable probabilistic system with N system states, in general;
- Calculate all probabilities introduced, i.e., p
_{1}, …, p_{N}.

## 3. Operational Complexity of Supplier-Customer Systems—Case Studies

- Case-collected data sheets are extracted from problem-oriented database properly either by structured query language (SQL) processing of reports generated by MIS, or manually, in the simplest case;
- Check all excerpted data for logical consistency;
- Statistical processing of the excerpted data and computation of entropy, e.g., issuing histograms (HIS), empirical distribution functions (EDF), and other additional numerical and/or graphical outputs, if necessary.

_{k}}, k = 1, .., K, which contains all observations available including repeating values, of a random variable Y. The Y makes a theoretical framework for any Q- or T-flow observed. The procedure has four basic steps:

- (i)
- Sort and scale {y
_{k}} by affine map in order to get {x_{k}} of a random variable X identically distributed as Y, but with dom(X) = [0, 1]. - (ii)
- Extract all repeating values form {x
_{k}} in order to get strictly increasing subset {x_{i}}, i = 1, …, N, 0 ≤ x_{1}< x_{2}< … < x_{N}≤ 1, with frequencies {f_{i}}, i = 1, ..., N of values which actually define system states. - (iii)
- Calculate EDF F(ξ) = P(ξ < x), x $\u03f5$ {x
_{i}}, with dom(F(.)) = range(F(.)) = [0, 1], and HIS, called an empirical frequency function alternatively, which gives relative frequencies {p_{i}} = {f_{i}/K}, i = 1, …, N. - (iv)
- Compute entropy and other related quantities, basically using Equation (2a,b) for calculation of H, and H
_{u}, or equivalents S and S_{u}based upon natural logarithms log(x) alternatively.

_{o}and receipt times T

_{d}of product deliveries. Hence, concerning the interface time quantities only, we may simplify denotations as follows:

- T
_{o}: order issue time, instead of_{i}_{,o}T_{i}, - T
_{d}: delivery time, instead of_{i}_{,d}T_{i},

_{i}is dropped as well, since it does not play a significant role in this analysis.

#### 3.1. Medium-Sized Building Engineering Company FA

_{u}are empirical distribution functions (EDF-s). The other parts of the presented figures show some raw data depicted in the form of continuous piecewise linear functions, other raw data with outliers purged, and relative frequencies, as well.

#### 3.2. Small-Sized Fashion Shop FB

#### 3.3. Medium-Sized Mechanical Engineering Company FC

#### 3.4. Small-Sized Lubricant Shop FD

_{u}upon lead-time tolerance thresholds in days, denoted [b

_{d}, b

_{u}], where b

_{d}denotes the lower bound, and b

_{u}the upper one, respectively. For a more detailed discussion and further details, we refer to [30].

_{u}= b, only. Surely, if we admit $\varphi $(b) to be time dependent on b, then h(b) = H(b)/H

_{u}(b) = H(b)/I

_{u}, will do as well.

_{u}is just a shorthand denotation of H

_{u}(b), used in the label of vertical axis in the Figure 14. The calculated values of h(b) are given in Table 7 and plotted in Figure 14.

#### 3.5. Top-Medium-Sized Mechanical Engineering Company FE

#### 3.6. Short Comparison of the Analyzed Study Cases

_{u}calculated for the particular firm and its commodity over all the firm’s suppliers. We are used to calling the argument of that particular minimization problem—the optimal supplier. They are listed in the fourth row, in particular for the firms FA, FB, and FC. In practice, these information can serve either for particular managerial decision making or to support a firm’s supplier negotiation process directly.

_{u}(b) upon an upper bound b of tolerance period [0, b], given in days. In particular, such an analysis is very important when finding a proper leverage between two firm’s aspects concerning any supplier—an acceptable tolerance in lead time variations of the supply stream versus its corresponding operation complexity measure. Because of the limited space here, we sketched this investigation of dependence of h(b) on b for one supplier, Sfd, and one commodity, Cfd, only. As far as the firm FE, we show in Figure 15, considering the limited space, too, just an illustration of complex structure of the raw input data stream of flow variations of the lead time collected from one supplier, Sfe, but for many commodities before sorting, during a period of four years.

## 4. Conclusions

_{u}to be a suitable, versatile and effective indicator thereto. However, one specific remark should be mentioned here, before closing this paragraph. In general, firms and their management in particular, are not inclined to provide detailed information about their supplier-customer relations, and not a bit of actual commodity flow variations in time and/or in volumes, at most. They include all such information in a set of strict firm and business secret data, which is quite natural and understandable. However in practice, it hinders or even precludes diffusing any such firm sensitive analysis results in public. Nevertheless, having selected five specific SME firms, we aimed to illustrate both the general steps of the proposed entropy-based procedure in detail, and some acceptable and specific results thereof, too. We hope other researchers will apply the discussed procedure for solving similar problems quite easily, if the data were obtained from practice.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Plot of functions ${S}_{c,d}^{*}\left(\varphi \right)$ and ${S}_{\text{BGS}}\left(\varphi \right)$ over (c,d) $\u03f5$ Ω for given $\varphi $ .

**Figure 2.**Curves representing the intersection ${S}_{c,d}^{*}\left(\varphi \right)$ $\cap$ ${S}_{\text{BGS}}\left(\varphi \right)$ over (c,d) $\u03f5$ Ω.

**Figure 4.**S1fA: concrete TrC 16–20, time gaps $\mathsf{\Delta}{T}_{d,o}$; (

**a**) EDF; (

**b**) values with outlier, continuous piecewise linear function.

**Figure 5.**S2fA: concrete TrC 16–20, time gaps $\mathsf{\Delta}{T}_{d,o}$; (

**a**) EDF; (

**b**) discrete values without outlier.

**Figure 6.**S1fA: solid brick CP 290 × 140 × 65, time gaps $\mathsf{\Delta}{T}_{d,o}$; (

**a**) EDF; (

**b**) empirical frequencies.

**Figure 7.**S2fA: Solid brick CP 290 × 140 × 65, time gaps $\mathsf{\Delta}{T}_{d,o}$; (

**a**) EDF; (

**b**) values with outlier, continuous piecewise linear function.

**Figure 8.**3-D bar plot of entropy ratios h of products (1—concrete, 2—solid brick, 3—building block) summarized by suppliers S1fA and S2fA laterally.

**Figure 9.**3-D bar plot of entropy ratios h of products (1—blouses, 2—dresses, 3—skirts) summarized by suppliers S1fB and S2fB laterally.

**Figure 10.**3-D bar plot of entropy ratios h of products (1—Tank G100, 2—Mast LTA, 3—Exchanger P12) summarized by suppliers S1fC and S2fC laterally.

**Figure 11.**$\mathsf{\Delta}{T}_{d,o}$ values; (

**a**) from unfiltered data, set [b

_{d}, b

_{u}] = [0, 0]; (

**b**) EDF.

**Figure 12.**$\mathsf{\Delta}{T}_{d,o}$ values; (

**a**) from filtered data, set [b

_{d}, b

_{u}] = [0, 7]; (

**b**) EDF.

**Figure 13.**$\mathsf{\Delta}{T}_{d,o}$ values; (

**a**) from filtered data, set [b

_{d}, b

_{u}] = [0, 14]; (

**b**) EDF.

**Figure 15.**Fractal character of $\mathsf{\Delta}{T}_{d,o}$ values plot containing all deliveries from 1 January 2007 till 31 December 2010.

0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
---|---|---|---|---|---|---|---|---|---|

103 | 113 | 86 | 118 | 94 | 101 | 95 | 88 | 101 | 101 |

p_{1} | p_{2} | p_{3} | p_{4} | p_{5} | p_{6} | p_{7} | p_{8} | p_{9} | p_{10} |
---|---|---|---|---|---|---|---|---|---|

0.103 | 0.113 | 0.086 | 0.118 | 0.094 | 0.101 | 0.095 | 0.088 | 0.101 | 0.101 |

– | – | Quantity | Time |
---|---|---|---|

(A) Supplier side | scheduled production | _{s,s}Q_{i}, i = 1, ..., n | _{s,s}T_{i}, i = 1, ..., n |

actual production | _{s,p}Q_{i}, i = 1, ..., n | _{s,p}T_{i}, i = 1, ..., n | |

(B) Interface | forecast | _{i}_{,f}Q_{i}, i = 1, ..., n | _{i,f}T_{i}, i = 1, ..., n |

order | _{i}_{,o}Q_{i}, i = 1, ..., n | _{i}_{,o}T_{i}, i = 1, ..., n | |

delivery | _{i}_{,d}Q_{i}, i = 1, ..., n | _{i}_{,d}T_{i}, i = 1, ..., n | |

(C) Customer side | scheduled production | _{c}_{,s}Q_{i}, i = 1, ..., n | _{c}_{,s}T_{i}, i = 1, ..., n |

actual production | _{c}_{,p}Q_{i}, i = 1, ..., n | _{c}_{,p}T_{i}, i = 1, ..., n |

**Table 4.**Building engineering company FA, suppliers S1fA and S2fA; values H, H

_{u}, calculated by Equation (2a,b), and ratios h = H/H

_{u}.

Supplier : Product | H | H_{u} | h = H/H_{u} |
---|---|---|---|

S1fA : Concrete | 2.55792 | 5.04439 | 0.507082 |

S2fA : Concrete | 2.66536 | 5.95420 | 0.447643 |

S1fA : Solid brick | 2.76999 | 5.08746 | 0.544474 |

S2fA : Solid brick | 2.62193 | 5.08746 | 0.515370 |

S1fA : Building block | 2.80140 | 4.52356 | 0.619292 |

S2fA : Building block | 2.93795 | 4.95420 | 0.593023 |

**Table 5.**Small-sized fashion shop FB, suppliers S1fB and S2fB; values H, H

_{u}, calculated by Equation (2a,b), and ratios h = H/H

_{u}.

Supplier : Product | H | H_{u} | h = H/H_{u} |
---|---|---|---|

S1fB : Blouses | 1.96692 | 4.00000 | 0.491729 |

S2fB : Blouses | 1.22791 | 6.04439 | 0.203148 |

S1fB : Dresses | 1.85475 | 4.45943 | 0.415917 |

S2fB : Dresses | 0.932112 | 4.52356 | 0.206057 |

S1fB : Skirts | 1.22791 | 6.04439 | 0.203148 |

S2fB : Skirts | 1.33920 | 6.37504 | 0.210069 |

**Table 6.**Medium-sized mechanical engineering company FC, suppliers S1fC and S2fC; values H, H

_{u}, calculated by Equation (2a,b), and ratios h = H/H

_{u}.

Supplier : Product | H | H_{u} | h = H/H_{u} |
---|---|---|---|

S1fC : Tank G100 | 2.31212 | 7.29462 | 0.316962 |

S2fC : Tank G100 | 2.69223 | 7.29462 | 0.369071 |

S1fC : Mast LTA | 3.24267 | 6.45943 | 0.502005 |

S2fC : Mast LTA | 3.34846 | 6.45943 | 0.518383 |

S1fC : Exchanger P12 | 2.31212 | 7.29462 | 0.316962 |

S2fC : Exchanger P12 | 2.52078 | 7.29462 | 0.345568 |

b | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 9 |

h(b) | 0.327 | 0.327 | 0.324 | 0.324 | 0.320 | 0.319 | 0.302 | 0.235 | 0.224 |

b | 10 | 11 | 12 | 13 | 14 | ||||

h(b) | 0.212 | 0.207 | 0.198 | 0.178 | 0.139 |

Study Case No. | 1 | 2 | 3 | 4 | 5 | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Firm | FA | FB | FC | FD | FE | ||||||

Commodity | C1fa | C2fa | C3fa | C1fb | C2fb | C3fb | C1fc | C2fc | C3fc | Cfd | many |

Optimal supplier | S2fa | S2fa | S2fa | S2fb | S2fb | S1fb | S1fc | S1fc | S1fc | Sfd | Sfe |

min h | 0.448 | 0.515 | 0.593 | 0.203 | 0.206 | 0.203 | 0.317 | 0.502 | 0.317 | 0.327 | - |

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons by Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

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Lukáš, L.; Hofman, J. Operational Complexity of Supplier-Customer Systems Measured by Entropy—Case Studies. *Entropy* **2016**, *18*, 137.
https://doi.org/10.3390/e18040137

**AMA Style**

Lukáš L, Hofman J. Operational Complexity of Supplier-Customer Systems Measured by Entropy—Case Studies. *Entropy*. 2016; 18(4):137.
https://doi.org/10.3390/e18040137

**Chicago/Turabian Style**

Lukáš, Ladislav, and Jiří Hofman. 2016. "Operational Complexity of Supplier-Customer Systems Measured by Entropy—Case Studies" *Entropy* 18, no. 4: 137.
https://doi.org/10.3390/e18040137