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entropy Article

Operational Complexity of Supplier-Customer Systems Measured by Entropy—Case Studies Ladislav Lukáš * and Jiˇrí Hofman Department of Economics and Quantitative Methods, Faculty of Economics, University of West Bohemia, Husova 11, Pilsen 30614, Czech Republic; [email protected] * Correspondence: [email protected]; Tel.: +420-377-633-107; Fax: +420-377-633-002 Academic Editors: J. A. Tenreiro Machado and António M. Lopes Received: 3 December 2015; Accepted: 28 March 2016; Published: 14 April 2016

Abstract: This paper discusses a unified entropy-based approach for the quantitative measurement of operational complexity of company supplier-customer relations. Classical Shannon entropy is utilized. Beside this quantification tool, we also explore the relations between Shannon entropy and (c,d)-entropy in more details. An analytic description of so called iso-quant curves is given, too. We present five case studies, albeit in an anonymous setting, describing various details of general procedures for measuring the operational complexity of supplier-customer systems. In general, we assume a problem-oriented database exists, which contains detailed records of all product forecasts, orders and deliveries both in quantity and time, scheduled and realized, too. Data processing detects important flow variations both in volumes and times, e.g., order—forecast, delivery—order, and actual production—scheduled one. The unifying quantity used for entropy computation is the time gap between actual delivery time and order issue time, which is nothing else but a lead time in inventory control models. After data consistency checks, histograms and empirical distribution functions are constructed. Finally, the entropy, information-theoretic measure of supplier-customer operational complexity, is calculated. Basic steps of the algorithm are mentioned briefly, too. Results of supplier-customer system analysis from selected Czech small and medium-sized enterprises (SMEs) are presented in various computational and managerial decision making details. An enterprise is ranked as SME one, if it has at most 250 employees and its turnover does not exceed 50 million USD per year, or its balance sheet total does not exceed 43 million USD per year, alternatively. Keywords: entropy; generalized entropy; information measure; operational complexity; supplier-consumer system

1. Introduction Business economics in general knows two types of complexity of supplier-customer systems. The first type is called structural complexity, which is defined as a static system variety and its main design dimensions. From an analytic point of view, it describes structural links among various business units and their hierarchies. It has dominantly a static representation and undergoes time changes usually over long-term periods. The second one is called operational complexity and concerns all uncertainties associated with system dynamics. From an analytic point of view, it reflects temporal changes in supplier-customer systems. In particular, an operational complexity measure should express the behavioral uncertainties of the system during time with respect to specified control levels. It has to record all possible types of flow variations within and across companies in detail, e.g., replenishment time disturbances, deviations of material in/out flows, etc. We assume that such data are available in a company management information system (MIS).

Entropy 2016, 18, 137; doi:10.3390/e18040137

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Finally, we would like to add that there are numerous types of complexity defined in various fields closely related to business economics. For example, there are three kinds of complexity within manufacturing systems—product, process and operational complexity, in particular. We refer to [1–5] for more details and various aspects of complexity including tools and methods applied in company management. From a management science point of view, a supplier-customer system belongs to the broad theory of inventory control [6]. There is well-known that links within any customer-supplier system can be represented by graphs as is usual in multi-echelon inventory systems. In this paper, the operational complexity of supplier-customer systems is analyzed using a unified entropy-oriented approach. We select five SME-ranked firms with rather diverse production lines from the West Bohemian district of the Czech Republic, and we present the results of our case studies in various details thus providing a general scheme for such type of managerial analyses. All these results are computed by entropy procedures based upon classical Shannon entropy. However, we also discuss a relation between the well-known Shannon entropy, or the Boltzmann–Gibbs–Shannon one as it is alternatively called (denoted as BGS-entropy, too), and a more general entropy, the (c,d)-entropy, in particular. The paper is organized as follows: in Section 2, we present a short theoretical background, which is divided into three parts: (i) the definition of entropy used to measure operational complexity, with some interesting results relating BGS-entropy and (c,d)-entropy; (ii) the information schemes of supplier-customer systems and the basic structure of the problem-oriented database; and (iii) a definition and qualitative description of the building of the variable records stored in such a particular database. In Section 3, which forms the core of the paper, we present the results of our five case studies. Some 3-D bar charts are added, too, which illustrate in a practical way how to recast entropy quantities into a form suitable for managerial decision making. In Section 4, we present a discussion of the results and some concluding remarks relating our future research. 2. Theoretical Background The theoretical framework for quantification of any system complexity is provided by information theory, in general. First, we may refer two seminal works on this field [7,8]. Entropy represents the most known quantitative measure of expected amount of information required to describe the state of a system, and builds a basic framework for the development of complexity theory [9,10]. Entropy is used for measuring the complexity of supply chains [11–17]. These papers bring manifold topics and approaches, and show also more or less connections with inventory control theory, in general. Another field of applications is focused on system complexity analysis, where there is engineering design and manufacturing chains, see [18–21]. In general, the complexity of a system increases with the increasing level of disorder and uncertainty of its states, as presented in [22]. A theoretical framework for joint conditional complexity is presented in [23]. Application of entropy for analysis and classification of numerical algorithms and the corresponding computational complexities is discussed in [24,25]. Paper [26] brings a good review of various multi-scale entropy algorithms and some applications, too. Two-echelon supply chain games and their information characteristics, in particular, are discussed in [27]. Finally, the papers [28–30] are concerned with measuring the operational complexity of supplier-customer systems, and initiated our serious interest in the field. An overview of basic Shannon–Khinchin axioms for entropy is given in [31,32], too, concluding that given a discrete probabilistic system, the only possible function which satisfies these axioms takes the following form: N ÿ (1) S pp1 , . . . , p N q “ ´c pi logb ppi q i “1

where c is a positive constant, c > 0, and base of logarithms is arbitrary b > 1.

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2.1. Entropy Used to Measure Operational Complexity Basically, we use classical Shannon information-theoretic measure and corresponding entropy defined for any information system with N states being characterized with a discrete probability ř distribution, i.e., each system state may appear with probability pi ě 0, with iN“1 pi “ 1. The Shannon entropy is given by following formula, which corresponds to Equation (1) taking c = 1 and b = 2: H pp1 , . . . , p N q “ ´

N ÿ

pi log2 ppi q

(2a)

i “1

where H is used to denote entropy calculated using log2 (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. The maximal value of H pp1 , . . . , p N q attainable is given for uniform distribution, and takes the form: ˆ ˙ 1 1 Hu :“ H p1 “ , . . . , p N “ “ log2 pNq (2b) N N Recently, generalized (c,d)-entropy was presented in [31], which is particularly interesting. It provides a versatile tool for quantitatively measuring complex systems, which are often inherently non-ergodic and non-Markovian ones, thus relaxing the rigorous need to apply Shannon entropy. Moreover, the (c,d)-entropy, with system-specific scaling exponents, c and d, has yet another attractive feature. This entropy contains many known entropy functionals as special cases, including Shannon entropy among others. Following [31], the (c,d)-entropy is defined as follows: Sc,d pp1 , . . . , p N q “ er

N ÿ

Γ pd ` 1, 1 ´ clog ppi qq ´ cr, r “ p1 ´ c ` cdq´1

(3)

i “1

using natural logarithmic function log(x), and incomplete gamma function Γ(a, b), as key-tools. The incomplete gamma function is given by: Γ pa, bq “

ż8

t a´1 exp ptq dt

(4)

b

when relaxing the lower integration bound of the Euler gamma function: Γ paq “

ż8

t a´1 exp ptq dt

(5)

0

which provides useful extension of the factorial, where (n ´ 1)! = Γ(n), for any positive integer n. The Boltzmann–Gibbs entropy originated in thermodynamics: SBGS pp1 , . . . , p N q “

N ÿ

pi log p1{pi q

(6)

i “1

It differs from Shannon entropy quantitatively just by using natural logarithms instead of logarithms with base 2. Hence, it is usually denoted by the symbol SBGS and called Boltzmann–Gibbs–Shannon entropy, in the most correct sense. Following [31], the relation between (c,d)-entropy and the Boltzmann–Gibbs–Shannon entropy is established for c = 1, and d = 1: SBGS pp1 , . . . , p N q “ S1,1 pp1 , . . . , p N q ´ 1

(7)

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This relation has motivated us to reformulate the additive term in Equation (3) in order to remove the additive constant ´1 from Equation (7) intrinsically. Hence, we propose the modified formula in the following form: Entropy 2016, 18, 137

˚ pp1 , . . . , p N q “ er Sc,d

N ÿ

4 of 17

Γ pd ` 1, 1 ´ clog ppi qq ´ c pr ` 1q , r “ p1 ´ c ` cdq´1

i “1

( ,…,

)=

∗ ,

( ,…,

)

(8)

(9)

and call it (c,d)-entropy as well, which still provides the desired smart relation:

Let denote the discrete distribution ( , … , ) of any given probabilistic system thus providing a way to write the next formulae and equations ˚ in a more compact way. pp1 , . . . , p N S1,1 ppof (9) SBGS . . . , p N q (c,d) over a region Ω ⊂ ℝ2, 1 , arguments Given , the ∗, ( ) can be investigated asqa“ function which helps significantly to elucidate Equations (7) and (9). Let φ denote discrete distribution probabilistic thus ® (versionpp10.1) N q of any 1 , . . .to, p perform our given numerical experimentssystem with We havethe used Mathematica providing a way to given write by theEquation next formulae in a more compact (c,d)-entropy (8), as and well equations as to discover a more general way. relation between ˚ SBGS than given by Equations (9) or (7), respectively. 2 (c,d)-entropy Given φ, the Sand c,d pφq can be investigated as a function of arguments (c,d) over a region Ω Ă R , 2 (c,d) Ω = to [0.1, 5] × [0.1,Equations 5] ⊂ ℝ , where denotes a Cartesian product of two sets. which helpsSet significantly elucidate (7) and× (9). The distribution is constructed from finite samples of random numbers generated by integer ® We have used Mathematica (version 10.1) to perform our numerical experiments with pseudo-random generators. In general, L will denote the length of list of such numbers, and their (c,d)-entropy given Equation (8), as range will be by {0, 1, …, kmax}, thus N =well kmax +as1.to discover a more general relation between (c,d)-entropy and SBGS than Equationsfunction (7) or (9), respectively. We given use theby Mathematica Random Integer [kmax, L], which serves exactly that purpose to a list integers from 0 toˆkmax desired.aSorting the listproduct for givenof kmaxtwo , Setgive (c,d) e of ΩL=pseudorandom [0.1, 5] ˆ [0.1, 5] ranging Ă R2 , where denotes Cartesian sets. the frequencies and corresponding empirical distribution will be calculated as an approximation The distribution φ is constructed from finite samples of random numbers generated by integer of a uniform distribution of probabilistic system with N states. pseudo-random generators. In general, L will denote the length of list of such numbers, and their We have run several numerical experiments varying both L and kmax. Since the results were very range will be {0, 1, . . . , k just }, thus N = kmaxones, + 1. see Figures 1 and 2. Raw data of the generated similar, we presentmax the following Weprobabilistic use the Mathematica function Random [kmax , L], serves exactly purpose to system are given in Table 1. In theInteger first row, there are which particular system states that selected, give a list pseudorandom ranging from N 0 to kmax therow, listthere for given kmax , i.e.,of tenLbins denoted 0, …, 9integers in sequel, which makes = 10 as kdesired. max = 9. In Sorting the second are the corresponding frequencies of state occurrences. Further, the be Table 2 gives the corresponding the frequencies and corresponding empirical distribution φ will calculated as an approximation of probabilities p1, …,ofp10probabilistic , and thus the distribution is easily a uniform distribution system with N states.obtained from Table 1. ( ) are plotted in Figure 1, where the blue horizontal plane The functions ∗, ( ) and We have run several numerical experiments varying both L and kmax . Since the results were ( ) to be constant over (c,d) Ω = [0.1, 5] × [0.1, 5], as it does not depend upon represents very similar, just the following ones, see Figures 1 and 2. Raw data of the generated (c,d) at we all present just by its definition. For given empirical distribution presented in Table 2, the probabilistic arewhich givenis in Table 1. In the row,asthere are particular ( system ) = 2.29809, a little bit lower thanfirst 2.30259, being calculated in casesystem of exact states uniformselected, i.e., ten distribution, bins denoted 0, . . . , 9 in sequel, which makes N = 10 as kmax = 9. In the second row, there precisely.

are the corresponding frequencies of state occurrences. Further, the Table 2 gives the corresponding Table 1. Raw data of probabilistic system sample generated by L = 1000, kmax = 9. probabilities p1 , . . . , p10 , and thus the distribution φ is easily obtained from Table 1. ˚1 pφq and 0 2 SBGS pφq 3 are plotted 4 5 Figure6 1, where 7 the blue 8 9 The functions Sc,d in horizontal plane 103 113 86 118 94 101 95 88 101 101 represents SBGS pφq to be constant over (c,d) e Ω = [0.1, 5] ˆ [0.1, 5], as it does not depend upon (c,d) at all just by its definition. For given empirical distribution φ presented in Table 2, the Table 2. Probabilities p1, …, p10 defining the probabilistic system distribution . SBGS pφq = 2.29809, which is a little bit lower than 2.30259, as being calculated in case of exact uniform 1 p2 p3 p4 p5 p6 p7 p8 p9 p10 distribution,pprecisely. 0.103

0.113

0.086

0.118

0.094

0.101

0.095

0.088

0.101

) and ( pφq ) over Figure 1. Plot functionsS˚∗, (pφq and SBGS (c,d) . φ. Figure 1. Plot of of functions over (c,d)Ωe for Ω given for given c,d

0.101

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Figure 2. Curves representing the intersection

∗ ,

( ) ∩

( ) over (c,d)

Ω.

˚ pφq X S Figure 2. Curves representing the intersection Sc,d BGS pφq over (c,d) e Ω.

The complicated surface of ∗, ( ) representing (c,d)-entropy being calculated for the same distribution within a sub-region (c,d) [1, 5] × [0.1, 0.75] is due to the numerical properties of Table 1. Raw datafunction of probabilistic system sample generated = 1000, result kmax has = 9. incomplete gamma calculations for the corresponding arguments. by ThisLsurprising ( ) over (c,d) Ω = [0.1, 5] × [0.1, 5] in issued a question to investigate an intersection ∗, ( ) ∩ more details. The intersection. 0 1 result is2given in 3Figure 2, 4which shows 5 the detected 6 7 8 9 In general, there are three curves representing the intersection of surface ∗, ( ) with the 103 plane 113 ( )86over (c,d) 118 Ω = [0.1, 94 5] × [0.1, 101 5], which 95 are plotted 88 in Figure 101 2. Given 101 , horizontal the analytic definition of set of points forming the intersection is the following: ∗ ( ) ( )} system distribution = , )defining ∈ Ω ⊂ ℝ | the (10)φ. Table 2. Probabilities p1 , .Λ.=. , (p10 , probabilistic

Provided Ω = Ω1 ∪ Ω2, we are able to localize the curves more distinctly. The mutually disjoint sub-regions Ω1p , Ω2, are defined using byptwo incidence p1 p3 p4 suitable p5 separating p6 linepgiven p9 points, p10 e.g., 2 7 8 (1,0) and (5,5), in the following way:

0.103

0.113

0.086

0.118

0.094

0.101

0.095

0.088

0.101

Ω = ( , ) ∈ Ω ⊂ ℝ | 4 − 5 ≥ −5}, Ω = ( , ) ∈ Ω ⊂ ℝ | 4 − 5 < −5

0.101

(11)

Hence, the sub-region Ω1 contains just one curve, denoted λ1, and called the main branch of the ˚ over representing ∗ ( ) The complicated surface (c,d)-entropy being calculated intersection, ∩of S(c,d) pφq (c,d) Ω, while sub-region Ω2 contains two other curves, , called secondary branches, on the contrary. The point (c,d) = (1,1), which stands in Equations (7) and distribution φ within a sub-region (c,d) e [1, 5] ˆ [0.1, 0.75] is due to the numerical (9) [31], thus belongs to λ1 evidently. Concluding, the curve λ1, i.e., the main branch, is given by:

for the same properties of incomplete gamma function calculations for the corresponding arguments. This surprising result has ( ), 4 − 5 ≥ −5} = ( , ) ∈ Ω ⊂ ℝ | ∗˚ (12) , ( )= pφq X SBGS pφq over (c,d) e Ω = [0.1, 5] ˆ [0.1, 5] in issued a question to investigate an intersection Sc,d Hence, given , the desired generalization of Equation (9) is the following: more details. The result is given in Figure 2, which shows the detected intersection. ˚ pφq with the ( ,⋯ ), ∀(intersection , )∈ , ) = ∗, ( , ⋯ , the In general, there are three curves representing of surface (13) Sc,d Of course, the secondary branches one[0.1, to express relations to Equation (13), 2. Given φ, the pφq over horizontal plane SBGS (c,d) e Ω = [0.1,allow 5] ˆ 5], which aresimilar plotted in Figure and could be also interesting, but we do not mention them in more details here. At the end of this analytic definition of we setmay of conclude points the forming the intersection is the following: section, relation between BGS-entropy and (c,d)-entropy is more complicated than a single-valued relation given just by (c,d) = (1,1), as in the Equations (9) and (7), respectively. ˚ Given the probability iso-quant curves on the Λ distribution, “ tpc, dq the P Ωvalue Ă Rof2 |BGS-entropy Sc,d pφq “induces SBGS the pφqu surface of (c,d)-entropy being calculated for the same probability distribution. The main branch of such iso-quant curves given by Equation (12), is that one which is incident with the (c,d) = (1,1) Ω =Ω point, exactly. 1 Y Ω2 , we are able to localize the curves more distinctly. The mutually

(10)

Provided disjoint sub-regions Ω1 ,2.2. ΩInformation using suitable separating line given by two incidence points, e.g., (1,0) 2 , are defined Scheme of Supplier-Customer Systems and Basic Structure of a Problem-Oriented Database and (5,5), in the following way: Basically, we expand herein upon our previous works on the topic [33–35], in particular. In order to analyze the operational complexity of supplier-customer systems, we have to define the

Ω1 “ tpc, dq P Ω Ă R2 | 4d ´ 5c ě ´5u, Ω2 “ tpc, dq P Ω Ă R2 | 4d ´ 5c ă ´5

(11)

Hence, the sub-region Ω1 contains just one curve, denoted λ1 , and called the main branch of the ˚ pφq X S intersection, Sc,d BGS pφq over (c,d) e Ω, while sub-region Ω2 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 λ1 evidently. Concluding, the curve λ1 , i.e., the main branch, is given by: ˚ λ1 “ tpc, dq P Ω Ă R2 | Sc,d pφq “ SBGS pφq , 4d ´ 5c ě ´5u

(12)

Hence, given φ, the desired generalization of Equation (9) is the following: ˚ pp1 , ¨ ¨ ¨ , p N q , @ pc, dq P λ1 SBGS pp1 , ¨ ¨ ¨ , p N q “ Sc,d

(13)

Of course, the secondary branches allow one to express relations similar to Equation (13), and could be also interesting, but we do not mention them in more details here. At the end of this section,

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we may conclude the relation between BGS-entropy and (c,d)-entropy is more complicated than a single-valued relation given just by (c,d) = (1,1), as in the Equations (9) and (7), respectively. Given the probability distribution, the value of BGS-entropy induces the iso-quant curves on the surface of (c,d)-entropy being calculated for the same probability distribution. The main branch of such iso-quant curves given by Equation (12), is that one which is incident with the (c,d) = (1,1) point, exactly. 2.2. Information Scheme of Supplier-Customer Systems and Basic Structure of a Problem-Oriented Database Basically, we expand herein upon our previous works on the topic [33–35], in particular. In order to analyze the operational complexity of supplier-customer systems, we have to define6the basic Entropy 2016, 18, 137 of 17 entities and corresponding flows of information, first. In general, we are able to identify three main basic entities and corresponding of information, first. Inwhere general, we are able toand identify three are entities—supplier, customer, and anflows interface between them, both supplier customer main entities—supplier, customer, and an interface between them, where both supplier and customer simply distinguishable as usual as being physically localizable uniquely. On the contrary, the are simply distinguishable as abstract usual as concept being physically localizable On localizable the contrary,much interface is more or less a good insomuch as beinguniquely. physically the interface is more or less a good abstract concept insomuch as being physically localizable much harder sometimes. harder sometimes. In this general framework, supplier and customer both schedule and realize their productions. In this general framework, supplier and customer both schedule and realize their productions. Thus, the operational complexity is measurable at the interface in particular, where quantities and times Thus, the operational complexity is measurable at the interface in particular, where quantities and of forecast, order, and actual aredeliveries detected.are Thedetected. basic information of a supplier-customer times of forecast, order,deliveries and actual The basic scheme information scheme of a system is sketched in Figure 3. supplier-customer system is sketched in Figure 3.

Figure 3. Information scheme of a supplier-customer system. Figure 3. Information scheme of a supplier-customer system.

In general, we assume that we are able to detect volumetric deviations in quantities of goods

In general, are able tosupply detecttimes volumetric deviations in quantities of goods delivered, aswe wellassume as, time that gaps we between actual and ordering ones, and other quantities delivered, well as,various time gaps between actual supply andRegistration ordering ones, and other quantities used toasmonitor replenishment deviations at thetimes Interface point. used to monitor various replenishment deviations at the Interface Registration point. 2.3. Definition of Variables

2.3. Definition Variablesa set of products {P1, ..., Pn} being handled within a supplier-customer system. Let usofconsider In general, there aare types of {P variables relating quantity and time to be considered for a Let us consider settwo of products 1 , ..., Pn } being handled within a supplier-customer system. particular product Pi, i = 1, ..., n. All of them are reported at both the supplier and customer side, and In general, there are two types of variables relating quantity and time to be considered for a particular thus, they should be reported at the interface, as well. Monitoring such variables provides time product Pi , i = 1, ..., n. All of them are reported at both the supplier and customer side, and thus, they series which form the core of information for measuring the operational complexity of should be reported at the interface, as well. Monitoring such variables provides time series which supplier-customer system. A list of typical variables monitored for products Pi, i = 1, ..., n, is given formin the core3.of information for measuring the operational complexity of supplier-customer system. Table A list of typical variables monitored for products Pi , i = 1, ..., n, is given in Table 3. Table 3. List of variables monitored for products Pi, i = 1, ..., n.

Table 3. List of variables monitored for products Pi , i = 1, ..., n. – – Quantity Time scheduled production s,sQi, i = 1, ..., n s,sTi, i = 1, ..., n (A) Supplier side – – Quantity Time actual production s,pQi, i = 1, ..., n s,pTi, i = 1, ..., n scheduled production ...,..., nn s,s Qi,fi ,Qi i= s,s Ti ,i,fiT=i, 1, , i 1, = 1, i =..., 1, n..., n forecast (A) Supplier side Q , i = 1, ..., n T , i = 1, ..., actual production s,p s,p i Qi, i = 1, ..., n i i,oTi, i = 1, n (B) Interface order i,o ..., n delivery i,d Q i , i = 1, ..., n i,d T i , i = 1, ..., n Q , i = 1, ..., n forecast T , i = 1, ..., n i,f i i,f i (B) Interface order i i= n n scheduled production , i1,= ..., 1, ..., i =..., 1, n..., n i ,o Qc,s i ,Q i ,o Ti ,c,siT=i, 1, (C) Customer side delivery i i= n n i,d Qc,p i ,Q i,d Ti ,c,piT=i, 1, actual production , i1,= ..., 1, ..., i =..., 1, n..., n scheduled production c,s Qi , i = 1, ..., n c,s Ti , i = 1, ..., n (C) Customer side Of course, depending upon the specific supplier-customer system actual production ..., n and collection ...,products n c,p Qi , i = 1, c,p Ti , i = 1,of handled along there, there might appear additional quantities, too. In general, we are able to define a lot of variables which are denoted systematically (a,b)Qi and (a,b)Ti, 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]. Since we are looking mainly for flow variations, we are to consider differences, e.g.,

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Of course, depending upon the specific supplier-customer system and collection of products handled along there, there might appear additional quantities, too. In general, we are able to define a lot of variables which are denoted systematically (a,b) Qi and (a,b) Ti , 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]. Since we are looking mainly for flow variations, we are to consider differences, e.g., (Order—Forecast), (Delivery—Order), (Actual production—Scheduled production), etc. Hence, such quantities are expressible in the general form: (a,b) Qi ´ (u,v) Qi , (a,b) Ti

´

(u,v) Ti ,

pa, bq ‰ pu, vq, a, u e ts, i, cu, b, v e ts, p, f, o, du

(14)

We assume that all of them should be kept in a problem-oriented database directly, or extractable and computable from available general manager information system (MIS) reports. However, natural and crucial questions arise—namely how to: ‚ ‚

Recast such data into a suitable probabilistic system with N system states, in general; Calculate all probabilities introduced, i.e., p1 , . . . , pN .

We know, the probability estimation depends on the specific supplier-customer system investigated. So far, from theoretical point of view, we conclude that all necessary probabilities will be estimated from the corresponding data collected by monitoring the supplier-customer system and stored in a proper problem-oriented database. 3. Operational Complexity of Supplier-Customer Systems—Case Studies General outline of data processing: ‚

‚ ‚

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.

The software package we have developed for numerical realization of this general procedure consists from several programs written in Java and Mathematica, respectively. The programs communicate each other by data file transfer using compatible data formats. The first step, as well as the second one, is performed by Java programs. The third step is performed by Mathematica notebooks exclusively. The EDF, entropy, as well as the other numerical characteristics, are calculated from raw data {yk }, 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 {yk } by affine map in order to get {xk } of a random variable X identically distributed as Y, but with dom(X) = [0, 1]. (ii) Extract all repeating values form {xk } in order to get strictly increasing subset {xi }, i = 1, . . . , N, 0 ď x1 < x2 < . . . < xN ď 1, with frequencies {fi }, i = 1, ..., N of values which actually define system states. (iii) Calculate EDF F(ξ) = P(ξ < x), x e {xi }, with dom(F(.)) = range(F(.)) = [0, 1], and HIS, called an empirical frequency function alternatively, which gives relative frequencies {pi } = {fi /K}, i = 1, . . . , N.

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(iv) Compute entropy and other related quantities, basically using Equation (2a,b) for calculation of H, and Hu , or equivalents S and Su based upon natural logarithms log(x) alternatively. For the paper, we have selected five SME-ranked firms from various branches and with different 8 of 17 and having their sites in the West Bohemian district of the Czech Republic. We denote these firms FA, FB, FC, FD, and FE, in sequence anonymously, so as to protect their business  To: order issue time, instead of i,oTi, secrets: FA—building engineering, FB—fashion shop, FC—mechanical engineering, FD—lubricant  Td: delivery time, instead of i,dTi, shop, and FE—transportation engineering. In a similar anonymous way, we denote suppliers, e.g., S1fA andthe S2fA, which denote two different of the company particular.i denoting the where, pre-index i denoting interfacesuppliers is dropped simply, and FA, the in post-index We are concerned with time flow variations exclusively, which are detected at the interface and product Pi is dropped as well, since it does not play a significant role in this analysis. express gaps between order timesorder To and receipt Thetime corresponding time gapissue between issue timetimes and Treceipt time isdeliveries. given by Hence, simple d of product concerning difference: the interface time quantities only, we may simplify denotations as follows: Entropy 2016, 18, 137 business orientations

− . To : order issue time, instead of i ,o Ti , ∆ , = (15) Td : delivery time, instead of i,d Ti , Such a quantity is rather important in practice. It is called lead time in inventory theory, and plays very significant role ininterface variousisinventory models.and In the thepost-index case studies selected, try to where,athe pre-index i denoting dropped simply, i denoting thewe product different aspects and variants of application entropy to measure the operational Pillustrate i is dropped as well, since it does not play a significant role in this analysis. complexity of supplier-customer systems. The corresponding time gap between order issue time and receipt time is given by ‚ ‚

simple difference: 3.1. Medium-Sized Building Engineering Company FA ∆Td,o “ Td ´ To . (15) The corresponding data for time gaps ∆ , calculation were collected from the year 2008 till Such a quantity is rather important in practice. It is called lead time in inventory theory, and 2010, and fetched into the problem-oriented database. Naturally, the company stores more details in plays a very significant role in various inventory models. In the case studies selected, we try to its own MIS database relating variations both in time and quantity of various products, but the most illustrate different aspects and variants of application entropy to measure the operational complexity important ones are—concrete, solid bricks, masonry mortars, plasters and building blocks. of supplier-customer systems. Figures 4–7 show some typical outputs from programs realizing the third step of our procedure, i.e., construction EDF-s,Engineering and other Company statistically 3.1. Medium-Sized of Building FA oriented graphs. Even though we only show the results of two products (concrete and solid brick), we can see how delicate would be to make The corresponding for time gaps ∆Td,o decision calculation were collected the year 2008 till S2fA, 2010, managerial decisions asdata to reach a preference between just twofrom suppliers S1fA and and fetched into the problem-oriented database. Naturally, the company stores more details in its own respectively. MIS database variations both inoftime and quantity of bricks variousbyproducts, thein most important The timerelating variations of deliveries concrete and solid supplierbut S1fA, particular the ones are—concrete, solid bricks, masonry mortars, plasters and building blocks. time gaps ∆ , , are depicted in Figures 4 and 5. The corresponding quantities of the second supplier 4–7 show typical outputs programs the third functions step of ourfor procedure, i.e., S2fAFigures are depicted insome Figures 6 and 7. Infrom all cases, the realizing most important calculating construction ofhEDF-s, statistically oriented graphs. Even though we parts only show results are other empirical distribution functions (EDF-s). The other of thethe presented entropy ratios = H/Huand of two products (concrete and solid brick), we can see how delicate would be to make managerial figures show some raw data depicted in the form of continuous piecewise linear functions, other raw decisions to reach a preference decision between just two suppliers S1fA and S2fA, respectively. data withas outliers purged, and relative frequencies, as well.

(a)

(b)

Figure 4. S1fA: concrete TrC 16–20, time gaps ∆ ; (a) EDF; (b) values with outlier, continuous Figure 4. S1fA: concrete TrC 16–20, time gaps ∆Td,o, ; (a) EDF; (b) values with outlier, continuous piecewise linear function. piecewise linear function.

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(a) (b) (a) (b) Figure 5. S2fA: concrete 16–20, time gaps ∆ , ; (a) EDF; (b) discrete values outlier. (a) TrC (b) without Figure 5. S2fA: concrete TrC 16–20, time gaps ∆T ; (a) EDF; (b) discrete values without outlier. Figure 5. S2fA: concrete TrC 16–20, time gaps ∆ Figure 5. S2fA: concrete TrC 16–20, time gaps ∆

,

,

d,o ; (a)

EDF; (b) discrete values without outlier. ; (a) EDF; (b) discrete values without outlier.

(a) (a) Figure 6. S1fA: solid brick (a) CP 290 × 140 × 65, time gaps ∆

,

(b) (b) ; (a) EDF; (b) empirical frequencies. (b)

Figure 6. S1fA: solid brick CP 290 × 140 × 65, time gaps ∆ , ; (a) EDF; (b) empirical frequencies. Figure 6. S1fA: solid brick CP 290 ˆ 140 ˆ 65, time gaps ∆T ; (a) EDF; (b) empirical frequencies. Figure 6. S1fA: solid brick CP 290 × 140 × 65, time gaps ∆ , ; d,o (a) EDF; (b) empirical frequencies.

(a) (a) Figure 7. S2fA: Solid(a) brick CP 290 × 140 × 65, time gaps ∆

,

(b) (b) ; (a) EDF; (b) (b) values with outlier,

Figure 7. S2fA: Solid linear brick function. CP 290 × 140 × 65, time gaps ∆ , ; (a) EDF; (b) values with outlier, continuous piecewise Figure 7. S2fA: Solid linear brick function. CP 290 × 140 × 65, time gaps ∆ ; (a) EDF; (b) values with outlier, continuous piecewise Figure 7. S2fA: Solid brick CP 290 ˆ 140 ˆ 65, time gaps ∆T,d,o ; (a) EDF; (b) values with outlier, continuous piecewise linear function. The corresponding results are summarized in Table 4 in a purely quantitative way, whilst in continuous piecewise linear function.

The8, corresponding results summarized in Table 4 ingiving a purely quantitative whilst in Figure the same results are are depicted graphically, thus a more intuitiveway, view of the The8, corresponding results are summarized in Table 4 ingiving a purely quantitative way, whilst in Figure the same results are depicted graphically, thus a more intuitive view of the performance comparison of both suppliers, i.e., S1fA and S2fA. Labels 1, 2, 3 subsequently denote The time variations of deliveries of concrete and solid bricks by supplier S1fA, in particular the Figure 8, the same results are depicted graphically, thus giving a more intuitive view of the performance comparison of bothi.e., suppliers, i.e., S1fA(2) and 1, 2,(3) 3 for subsequently denote products in the frontal direction, (1) for concrete; forS2fA. solid Labels brick; and building block, in time gaps ∆T , are depicted in Figures 4 and 5. The corresponding quantities of the second supplier performance comparison of bothi.e., suppliers, i.e., S1fA(2) and S2fA. 1, 2,(3) 3 for subsequently denote d,o products inThe the frontal direction, for concrete; for solid Labels brick;decision and building block, in particular. main purpose of such(1) results is to support managerial making. S2fA are depicted in Figures 6 and 7. In(1) allfor cases, the most important functions for building calculating entropy products inThe themain frontal direction, i.e., concrete; (2) for solid brick; and (3) making. block, in particular. purpose of such results is to support managerial decision In the present case of building engineering firm FA, the supplier preference question seems to ratios hIn = H/H are empirical distribution functions (EDF-s). The other parts of the presented figures particular. The main purpose of such results is to support managerial decision making. u the present of building firm FA, the which supplier preference question decision. seems to be answered easily.case It is evident thatengineering S2fA outperforms S1fA, is the clear managerial In theraw present case of building engineering firm FA, the which supplier preference question seems to show some data depicted in the form of continuous piecewise linear functions, other raw data be answered easily. It is evident that S2fA outperforms S1fA, is the clear managerial decision. Of course, keeping in mind that we have analyzed time gaps ∆ , , only. be answered easily. It is evident that S2fA outperforms S1fA, which is the clear managerial decision. with purged,in and relative frequencies, as well. Ofoutliers course, keeping mind that we have analyzed time gaps ∆ , , only. OfThe course, keeping in mind thatare wesummarized have analyzedintime gaps ∆ a , purely , only. quantitative way, whilst in corresponding results Table 4 in Figure 8, the same results are depicted graphically, thus giving a more intuitive view of the performance comparison of both suppliers, i.e., S1fA and S2fA. Labels 1, 2, 3 subsequently denote products in the

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Table 4. Building engineering company FA, suppliers S1fA and S2fA; values H, Hu, calculated by Equation (2a,b), and ratios h = H/Hu. Entropy 2016, 18, 137

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h = H/Hu Supplier : Product H Hu S1fA : Concrete 2.55792 5.04439 0.507082 frontal direction, i.e., (1) for: Concrete concrete; (2) for solid brick; and (3) for building S2fA 2.66536 5.95420 0.447643block, in particular. The main purpose of such results is to support managerial decision making. S1fA : Solid brick 2.76999 5.08746 0.544474 In the present case of building engineering firm FA, the supplier preference S2fA : Solid brick 2.62193 5.08746 0.515370 question seems to be answered easily. S1fA It is evident that S2fA outperforms S1fA, which is the clear managerial decision. : Building block 2.80140 4.52356 0.619292 Of course, keeping in mind that we have analyzed time gaps ∆T , only. d,o S2fA : Building block 2.93795 4.95420 0.593023

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

3.2. Small-Sized Fashion Shop FB

Table 4. Building engineering company FA, suppliers S1fA and S2fA; values H, Hu , calculated by Equation (2a,b), and ratios h = H/H The corresponding analysis in this u . case is almost similar to the previous one, i.e., company FA.

However, the product line is quite different one, i.e., fashion goods, and moreover, the size of the Supplier : Product Hu h = H/Hu company FB is much smaller than the size of FA.H Again, we investigated role of two main suppliers, denoted0.507082 S1fB, and S2fB, now, under the S1fAthe : Concrete 2.55792 5.04439 S2fA : Concrete 0.447643 objective of lead time disorder measured by 2.66536 quantity ∆5.95420 , , too. The most important commodities 5.08746 : Solid brick for FB in the specified S1fA season are—blouses,2.76999 dresses, and skirts. 0.544474 We present just final results in S2fA : Solid brick 2.62193 5.08746 0.515370 Table 5 and Figure 9. S1fA We :can see that making process need not be as simple as in the Building blockdecision 2.80140 4.52356 0.619292 previous case. However, the: Building management concluded that the 0.593023 supplier S2fB is the right choice, S2fA block of FB 2.93795 4.95420 even though her outperformance the concurrent supplier S1fB is strictly achieved in blouses and dresses. On theFashion contrary, 3.2. Small-Sized Shopthe FB performance of supplier S1fB is slightly better being compared to supplier S2fB in the commodity of skirts. Thus, we have reached two different conclusions. In such a The corresponding case is almost similarmulti-objective to the previous choice one, i.e.,ofcompany situation, one can eitheranalysis utilize in anthis algorithm of weighted variants,FA. in However, the product line is quite different one, i.e., fashion goods, and moreover, the size of the general, or apply another selection algorithm, e.g., based simply upon preferring previous good company is much smaller than the sizeAs of FA. experienceFBwith a particular supplier. we have already said, the management of FB has Again, we investigated the role of two main suppliers, S1fB, andsupplier S2fB, now, the negotiated a business contract with supplier S2FB to bedenoted their exclusive of under the three objective of lead time disorder measured by quantity ∆T , too. The most important commodities for d,o commodities considered for the next period of time. FB in the specified season are—blouses, dresses, and skirts. We present just final results in Table 5 and Figure 9. We can see that decision process need be asvalues simple previous Table 5. Small-sized fashion shop making FB, suppliers S1fB andnot S2fB; H,asHin u, the calculated by case. However, the management of FB concluded that the supplier S2fB is the right choice, even though Equation (2a,b), and ratios h = H/Hu. her outperformance the concurrent supplier S1fB is strictly achieved in blouses and dresses. On the h = H/Hu Supplier : Product H Hu contrary, the performance of supplier S1fB is slightly better being compared to supplier S2fB in the S1fB : Blouses 1.96692 4.00000 0.491729 commodity of skirts. Thus, we have reached two different conclusions. In such a situation, one can S2fBof: Blouses 1.22791 6.04439 0.203148 either utilize an algorithm weighted multi-objective choice of variants, in general, or apply another S1fB : Dresses 1.85475 0.415917 selection algorithm, e.g., based simply upon preferring 4.45943 previous good experience with a particular : Dresses 0.932112 of 4.52356 0.206057 supplier. As we have S2fB already said, the management FB has negotiated a business contract with supplier S2FB to be their exclusive supplier of the three commodities considered for the next period of time.

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S1fB : Skirts S2fB : Skirts

1.22791 1.33920

6.04439 6.37504

0.203148 0.210069

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

3.3. Medium-Sized Mechanical Engineering Company FC

Table 5. Small-sized fashion shop FB, suppliers S1fB and S2fB; values H, Hu , calculated by Equationthe (2a,b), and ratiosofh the = H/H Again, basic steps corresponding analysis are almost similar to those ones done for u.

the company FA. However, the product line is a rather specific one, i.e., production of Supplier : Product for assembling H H h =mainly H/Hu civil engineering ones. u purpose-oriented mechanical components structures, The most important products FC are—tanks, masts, and heat-exchangers. S1fB for : Blouses 1.96692 4.00000 0.491729 S2fBpresentations : Blouses 6.04439 Keeping the line of case as1.22791 compact as possible, 0.203148 we analyze the role of two main 1.85475 4.45943 0.415917 suppliers, denoted S1fC, S1fB and: Dresses S2fC. We apply the objective of lead time disorder measured by 4.52356 0.206057 S2fB : Dresses 0.932112 quantity ∆ , , again. Final results study are presented in Table 6 and Figure 10, in S1fB : Skirtsof this case 1.22791 6.04439 0.203148 particular. Concluding both quantitative results and graphical ones, we can see the clear result—a S2fB : Skirts 1.33920 6.37504 0.210069 monotone outperformance of the supplier S1fC in all three products analyzed, albeit in a relatively weak sense. 3.3. Medium-Sized Mechanical Engineering Company FC Again,6.the basic steps mechanical of the corresponding similar to those done Table Medium-sized engineering analysis companyare FC,almost suppliers S1fC and S2fC;ones values H, Hfor u, the company FA. However, the product line is a rather specific one, i.e., production of purpose-oriented calculated by Equation (2a,b), and ratios h = H/Hu. mechanical components for assembling structures, mainly civil engineering ones. The most important h = H/Hu Supplier : Product H Hu products for FC are—tanks, masts, and heat-exchangers. S1fC : Tank G100 2.31212 7.29462 0.316962 Keeping the line of case presentations as compact as possible, we analyze the role of two main S2fC : Tank 2.69223 0.369071 suppliers, denoted S1fC, and S2fC.G100 We apply the objective of7.29462 lead time disorder measured by quantity S1fC :ofMast 6.45943 ∆Td,o , again. Final results this LTA case study 3.24267 are presented in Table 6 0.502005 and Figure 10, in particular. S2fC : Mast LTAand graphical 3.34846ones, 6.45943 0.518383 Concluding both quantitative results we can see the clear result—a monotone S1fC : Exchanger P12 2.31212 7.29462 0.316962 outperformance of the supplier S1fC in all three products analyzed, albeit in a relatively weak sense. S2fC : Exchanger P12 2.52078 7.29462 0.345568 Table 6. Medium-sized mechanical engineering company FC, suppliers S1fC and S2fC; values H, Hu , calculated by Equation (2a,b), and ratios h = H/Hu . Supplier : Product

H

Hu

h = H/Hu

S1fC : Tank G100 S2fC : Tank G100 S1fC : Mast LTA S2fC : Mast LTA S1fC : Exchanger P12 S2fC : Exchanger P12

2.31212 2.69223 3.24267 3.34846 2.31212 2.52078

7.29462 7.29462 6.45943 6.45943 7.29462 7.29462

0.316962 0.369071 0.502005 0.518383 0.316962 0.345568

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Figure 10. 3-D bar plot of entropy ratios h of products (1—Tank G100, 2—Mast LTA, 3—Exchanger Figure 10. 3-D bar plot of entropy ratios h of products (1—Tank G100, 2—Mast LTA, 3—Exchanger P12) P12) summarized by suppliers S1fC and S2fC laterally. summarized by suppliers S1fC and S2fC laterally. Figure 10. 3-D bar plot of entropy ratios h of products (1—Tank G100, 2—Mast LTA, 3—Exchanger P12) summarized by Shop suppliers 3.4. Small-Sized Lubricant FD S1fC and S2fC laterally.

3.4. Small-Sized Lubricant Shop FD Contrary to fashion shop FB that sells season-dependent commodities, the lubricant shop FD 3.4.Contrary Small-Sized Shop FB FD to Lubricant fashion shop that sells season-dependent commodities, the lubricant shop FD largely sells season-invariant products. The company has a dominant supplier whose behavior can largely Contrary sells season-invariant products. Theseason-dependent company has a commodities, dominant supplier whoseshop behavior to fashion shop FB that sells the lubricant FD cause serious problems by not-fulfilling the settled contracts, so we have primarily considered season-invariant The company has a dominant supplier behavior can canlargely cause sells serious problems byproducts. not-fulfilling the settled contracts, so we have whose primarily considered deviations ∆ , , again. The problem-oriented database consists of 452 data covering orders and cause serious by problem-oriented not-fulfilling the settled contracts, so of we452 have primarily deviations ∆Td,o ,problems again. The database consists data coveringconsidered orders and deliveries of goods over the whole year 2011. The main goal of the analysis is to show the time deviations ∆ , again. The problem-oriented database consists of 452 data covering orders deliveries of goods over the whole year 2011. The main goal of the analysis is to show theand time , upon lead-time tolerance thresholds in days, denoted [b d, bu], dependence of goods entropy ratio h = H/Huyear deliveries of the 2011. The main goal ofthresholds the analysis is to show the [b time dependence of entropyover ratio h =whole H/Hu upon lead-time tolerance in days, denoted d , bu ], where bd denotes the lower bound, andu upon bu the upper one, respectively. For ainmore tolerance thresholds days,detailed denoteddiscussion [bd, bu], dependence of entropy ratio h = H/H where bd denotes the lower bound, and bu thelead-time upper one, respectively. For a more detailed discussion and further details, we refer to [30]. where b d denotes the lower bound, and bu the upper one, respectively. For a more detailed discussion and further details, we refer to [30]. Inspecting the results presented in Figure 11, we may observe a clustering of deviations at 7-day and further details, we refer to [30]. in Inspecting the results presented Figure 11, we may observe a clustering of deviations at 7-day long periods, means in practice delays show of week-long Inspectingwhich the results presented in Figurethat 11, wedelivery may observe a clustering deviations anomalies at 7-day long periods, which means in practice that delivery delays show week-long anomalies pre-dominantly. pre-dominantly. is supported results presented in Figures 12 and 13. week-long Finally, the dependence long periods, This which means inbypractice that delivery delays show anomalies This is supported by results presented in Figures 12 and 13. Finally, the dependence of EDF-s provides of pre-dominantly. EDF-s provides This evidence that theby probability distribution of lead time,13.denoted , again, shows a is supported results presented in Figures 12 and Finally, the dependence evidence that the probability distribution of lead time, denoted φ, again, shows a dependence upon dependence upon variable threshold periods [0, distribution b], with a variable upperdenoted bound bu, =again, b, only. Surely, of EDF-s provides evidence that the probability of lead time, shows a variable threshold periods [0, b], with a variable upper bound bu = b, only. Surely, if we admit φ(b) to dependence upon variable threshold periods [0, b], with a variable u =as b, well. only. Surely, if we admit (b) to be time dependent on b, then h(b) = H(b)/H u(b) =upper H(b)/Ibound u, will b do be time dependent on b, then h(b) = H(b)/Hu (b) = H(b)/Iu , will do as well. if we admit (b) to be time dependent on b,ofthen h(b)used = H(b)/H u(b) = H(b)/I u, will do as well. Note: Iu is just a shorthand denotation Hu(b), in the label of vertical axis in the Figure 14. Note: Iu Iis just a ashorthand denotation of H (b),used usedininthe thelabel labelofofvertical vertical axis in Figure u(b), Note: u is just shorthand denotation of H u axis in thethe Figure 14.14. The calculated values of h(b) are given in Table 7 and plotted in Figure 14. TheThe calculated values ofofh(b) Figure 14. 14. calculated values h(b)are aregiven givenininTable Table77and and plotted plotted in in Figure

(a) (a)

(b) (b)

Figure data,set set[b[bd,d,bbu]u]==[0, [0,0];0];(b) (b)EDF. EDF. Figure11. 11.∆∆ , values; values;(a) (a)from from unfiltered unfiltered data, Figure 11. ∆Td,o, values; (a) from unfiltered data, set [bd , bu ] = [0, 0]; (b) EDF.

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(a) (b) (a) (b) Figure (a) 12. ∆ , values; (a) from filtered data, set [bd, bu] = [0, 7];(b) (b) EDF. Figure 12. ∆T values; (a) from filtered data, set [b , b ] = [0, EDF. Figure 12. d,o ∆ , values; (a) from filtered data, set [bdd, bu]u= [0, 7]; 7]; (b) (b) EDF. Figure 12. ∆

,

values; (a) from filtered data, set [bd, bu] = [0, 7]; (b) EDF.

(a) (b) (a) (b) Figure 13. (a) ∆ , values; (a) from filtered data, set [bd, bu] = [0, 14]; (b)(b) EDF. Figure 13. ∆ , values; (a) from filtered data, set [bd, bu] = [0, 14]; (b) EDF. Figure 13. ∆Td,o values; (a) from filtered data, set [bd , bu ] = [0, 14]; (b) EDF. Figure 13. ∆

,

values; (a) from filtered data, set [bd, bu] = [0, 14]; (b) EDF.

Figure 14. Entropy ratio h(b) = H(b)/Hu(b) from data filtered by [0, b], b = 0, 1, …,14 days. Figure 14. Entropy ratio h(b) = H(b)/Hu(b) from data filtered by [0, b], b = 0, 1, …,14 days. Figure 14. Entropy ratio = H(b)/H u(b) from data filtered by [0, b], b = 0, 1, …,14 days. Table 7. h(b) Entropy ratio h(b) vs. tolerance upperby bound Figure 14. Entropy ratio h(b) = H(b)/H from data filtered [0, b],b.b = 0, 1, . . . ,14 days. u (b) Table 7. Entropy ratio h(b) vs. tolerance upper bound b. b 0 1 2 4 5 bound b.6 7 9 Table 7. Entropy ratio 3 h(b) vs. tolerance upper

b 0 1 7. Entropy 2 ratio0.324 3 vs. tolerance 4 5 bound 6b. 7 9 Table h(b) upper h(b) 0.327 0.327 0.324 0.320 0.319 0.302 0.235 0.224 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) 0.327 0.327 0.324 0.324 0.320 0.319 bb 10 11 0 1 212 313 414 5 60.302 70.235 90.224 h(b) 0.212 0.207 0.198 0.178 0.139 b 10 11 12 13 14 h(b) 0.327 0.327 0.324 0.324 0.320 0.319 0.302 0.235 0.224 h(b) 0.212 0.207 0.198 0.178 0.139 h(b) 0.212 0.207 0.178 0.139 From point of 0.198 view, such b a managerial 10 11 12 13 results,14as in Figures 8–10, are very important ones From a managerial point of view, such results, asand in Figures 8–10,ofareproper very important ones h(b)they 0.212 0.207 0.198 for 0.178 0.139 because can be directly used negotiation settlement details within Fromthey a managerial point ofused view,for such results, asand in Figures 8–10,ofareproper very important ones because can be directly negotiation settlement details within supplier-consumer contracts. Another field of implementation occurs in problems related to because they can be directlyAnother used for negotiation and settlement ofinproper details within supplier-consumer field of implementation sensitivity analysis ofcontracts. various management decisions in logistics andoccurs inventoryproblems control. related to From a managerial point of view, suchfield results, in are very importantrelated ones because supplier-consumer contracts. Another of as implementation in problems to sensitivity analysis of various management decisions in Figures logistics 8–10 andoccurs inventory control. they can be directly used for negotiation anddecisions settlement of proper within supplier-consumer sensitivity analysis of various management in logistics anddetails inventory control.

contracts. Another field of implementation occurs in problems related to sensitivity analysis of various management decisions in logistics and inventory control.

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

This company is the biggest one among all the discussed ones, and identified as belonging in This company is the biggest one among all the discussed ones, and identified as belonging in the top SME size-rank. The main production line of FE consists of transportation vehicles for public the top SME size-rank. The main production line of FE consists of transportation vehicles for public transport, e.g., trolleybuses, electric locomotives, and others. The company has its own MIS DB. transport, e.g., trolleybuses, electric locomotives, and others. The company has its own MIS DB. Corresponding datadata werewere excerpted from a huge MS-Excel file,file, which was generated byby a MIS report Corresponding excerpted from a huge MS-Excel which was generated a MIS generator. It contains more than 42,000 records in total. The company management was interested report generator. It contains more than 42,000 records in total. The company management was in lead time deviations products, irrelevant of which type, i.e., tangible or in-tangible interested in leadacross time all deviations across all products, irrelevant of which type, i.e., tangibleones, or as ones, as immense, well. The analysis immense,of and the discussion of results, too,show so here, well. in-tangible The analysis was and thewas discussion results, too, so here, we just thewe fractal just show thelead fractal character of the plot, lead time deviations plot, i.e., ∆ values, depicted in Figure character of the time deviations i.e., ∆T values, depicted in Figure 15, using a day as , d,o 15, using a day as a natural unit for both horizontal and vertical axes. a natural unit for both horizontal and vertical axes.

Figure 15. Fractal character of ∆

values plot containing all deliveries from 1 January 2007 till 31

Figure 15. Fractal character of ∆Td,o, values plot containing all deliveries from 1 January 2007 till December 2010. 31 December 2010.

3.6. Short Comparison of the Analyzed Study Cases

3.6. Short Comparison of the Analyzed Study Cases

Although the five selected firms are very different ones not only as regarding their business

orientation,the but five also selected their product lines, suppliers, etc.,only apartas from the pointtheir of view of Although firms aremanagement, very different ones not regarding business their enterprise range. In general, they fit all into the SME range in accordance with their staff orientation, but also their product lines, management, suppliers, etc., apart from the point of view headcount and range. balance In sheet total, which as into it is well-known, the main two factors with used for an staff of their enterprise general, they are, fit all the SME range in accordance their enterprise range determination. We denoted the firms FA, FB, …, FE thus maintaining their headcount and balance sheet total, which are, as it is well-known, the main two factors used for an anonymity. Following a simple sub-coordination principle, their suppliers are denoted Sjfz enterprise range determination. We denoted the firms FA, FB, . . . , FE thus maintaining their anonymity. accordingly. Here, j gives nothing else but a formal order number within a list of particular firm’s Following a simple sub-coordination principle, their suppliers are denoted Sjfz accordingly. Here, j accepted suppliers, e.g., j = 1, 2, in case of two suppliers only, and z ranges {a, b, …, e}, which allows gives one nothing else but a formal orderThe number within are a list of particular firm’s accepted suppliers, to uniquely identify the firm. commodities denoted in a similar fashion, i.e., Cjfz, with j e.g., j = 1, and 2, inzcase of two suppliers only, and z above. ranges {a, b, . . . , e}, which allows one to uniquely identify having the same meaning as given the firm. The commodities are calculations denoted inare a similar fashion, i.e., Cjfz, j and z having the same The main results of our clearly given in Table 8. Thewith last two rows are the most important therein. The fifth row gives the minimal values of entropy ratio h = H/H u calculated for the meaning as given above. particular firm and its commodity over all the firm’s suppliers. We are used to calling the argument The main results of our calculations are clearly given in Table 8. The last two rows are the most of thattherein. particular problem—the optimal supplier. They are listed the fourth row, in important Theminimization fifth row gives the minimal values of entropy ratio h = in H/H u calculated for the particular for the firms FA, FB, and FC. In practice, these information can serve either for particular particular firm and its commodity over all the firm’s suppliers. We are used to calling the argument managerial decision making or to support a firm’s supplier negotiation process directly. 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.

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Table 8. The optimal suppliers and their minimal values of entropy ratio h = H/Hu . Study Case No.

1

2

3

4

5

Firm

FA

FB

FC

FD

FE

Cfd Sfd 0.327

many Sfe -

Commodity Optimal supplier min h

C1fa S2fa 0.448

C2fa S2fa 0.515

C3fa S2fa 0.593

C1fb S2fb 0.203

C2fb S2fb 0.206

C3fb S1fb 0.203

C1fc S1fc 0.317

C2fc S1fc 0.502

C3fc S1fc 0.317

However, the study cases of firms FD and FE and their results are a little bit different ones. The main goal of operational complexity analysis within the firm FD was to show an investigation of dependence of entropy ratio h(b) = H(b)/Hu (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. Finally, we have to emphasize that we have analyzed flow variations of lead time ∆Td,o , only, to keep in line with our specific motivation consistency having been accepted intentionally. However, the same procedure can be used for analyses of flow variations of other quantities, too. In general, flow variations of different volumetric quantities are recommended be converted into dimensionless ones simply by some norm or rescaling processes with suitably selected denominators first, before submitting them to processing for complexity analysis by entropy-based procedures. 4. Conclusions The measurement of operational complexity based upon entropy provides a versatile instrument for supplier-customer system analysis in practice, as well as motivations for theoretical research. In this paper, we have presented not only the well-known Shannon entropy, or Boltzmann–Gibbs–Shannon as it is alternatively called, but also a more general approach based upon the (c,d)-entropy. Our main contribution in this field presented in this paper is an investigation of the relation between BGS-entropy and (c,d)-entropy. We have shown it is more complicated than a single-valued relation given just by (c,d) = (1,1). Given the probability distribution, the value of BGS-entropy induces iso-quant curves on the surface of (c,d)-entropy being calculated for the same probability distribution. To our best knowledge, this is a new finding. In the case studies presented, we applied a unified approach for the analysis of lead time variations of product-line main products at five SME-ranked firms. The results are briefly discussed, and possibilities of their direct application for managerial decision making are briefly outlined too. However, the firms are rather diverse from each other. They are not only of different size, but they all differ in their primary production lines. Their suppliers are different, too. Hence, we think that each case study should be concluded separately. The main purpose of all case studies presented, and we hope our modest contribution in the field of measuring operational complexity of supplier-customer systems, is the clear promotion of entropy ratio h = H/Hu 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

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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. We doubt, and we were even not able to get corresponding data describing flow variations of commodity deliveries from some typical and internationally renowned companies. Another question is to generate some sample streams of such data by random simulations, which could be distributed for public use, thus providing a chance to run fully reproducible entropy calculation benchmarks. However, it was not our desire, here. Evidently, from both the managerial and theoretical point of view there is still a lot of work to do. In particular, collection and processing of data, probability estimation of all mutually disjunctive states in specific supplier-customer system considered, and last but not least an accumulation of experience with different kinds of applications. Another interesting topic, which is to be investigated, too, is linked with the complexity analysis and computation of particular entropy measures of information sets containing deviations of various time and other quantities flows simultaneously. One idea is based upon proper rescaling of different deviation flows in order to get dimensionless quantity flows exclusively. The other one tries to utilize various generalized entropy approaches and algorithms thereon. Our research is still ongoing in a rather broad field of entropy-based measurement of the operational complexity of supplier-customer systems both in theory and practical applications. Acknowledgments: We gratefully acknowledge the assistance of comments and suggestions from three anonymous referees. The research project was supported by the Grant No. 15-20405S of the Grant Agency, Prague, Czech Republic. Author Contributions: Ladislav Lukáš conceived the approach, designed and performed the numerical computations and wrote the initial version of the manuscript. Jiˇrí Hofman collected the data used in our case studies. Both authors have read and approved the final manuscript. Conflicts of Interest: The authors declare no conflict of interest.

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