Efficiency of activities in production networks of arbitrary structures and technologies

Dyckhoff, Harald; Souren, Rainer Article — Published Version Efficiency of activities in production networks of arbitrary structures and technologies Journal of Business Economics Provided in Cooperation with: Springer Nature Suggested Citation: Dyckhoff, Harald; Souren, Rainer (2025) : Efficiency of activities in production networks of arbitrary structures and technologies, Journal of Business Economics, ISSN 1861-8928, Springer, Berlin, Heidelberg, Vol. 95, Iss. 6, pp. 809-838, https://doi.org/10.1007/s11573-025-01228-9 This Version is available at: https://hdl.handle.net/10419/330401 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/ Vol.:(0123456789) Journal of Business Economics (2025) 95:809–838 https://doi.org/10.1007/s11573-025-01228-9 ORIGINAL PAPER Efficiency ofactivities inproduction networks ofarbitrary structures andtechnologies HaraldDyckhoff1 · RainerSouren2 Accepted: 17 February 2025 / Published online: 17 March 2025 © The Author(s) 2025 Abstract Production takes place in complex networks. An important question is how the efficiency of the whole network is related to that of the individual production units forming the network. A general network production theory with following characteristics is developed. Analysed networks may possess arbitrary structures with units whose technologies may be non-convex and even discrete. The theory generalises Koopmans’ linear activity analysis based on similar underlying modelling features and fundamental assumptions. The modelling approach is more suitable for analysing networks than common ones. Methods of efficiency measurement known from network data envelopment analysis are integrated into the theory. It is shown that calculating an overall efficiency score for a network as average of individual scores of its units is inappropriate. The relationship between the efficiency of a network activity and that of subsystems and units strongly depends on the extent of which the individual production units are free to choose their input and output quantities, i.e. whether the network is loose or tied. Especially in cases where flows of intermediate products are constrained instead of freely disposable, the explicit modelling of their overproduction helps to analyse their influence on efficiency scores. Keywords Activity analysis· Production networks· Efficiency analysis· Nonconvex technology· Network DEA JEL Classification C14· C67· D24· L25 * Harald Dyckhoff Dyc[email protected]h-aachen.de Rainer Souren Rainer[email protected] 1 School ofBusiness andEconomics (Former Chair ofBusiness Theory, Sustainable Production, andIndustrial Control), RWTH Aachen Technical University, Templergraben 64, 52056Aachen, Germany 2 Group ofSustainable Production andLogistics Management, Ilmenau University ofTechnology, Postbox100565, 98684Ilmenau, Germany 810 H.Dyckhoff, R.Souren 1 Introduction Production is a human-directed and controlled process that uses and transforms selected objects (including services) as input to create new objects that emerge as output from the process. It is aimed at value creation, i.e., total advantages generated shall outweigh total disadvantages. Usually, the transformation takes place in systems consisting of several units that produce largely independently from each other, except for being linked by objects that are delivered as output from one unit and received as input by another unit within the system. Such a system may be a machine factory with working stations, a chemical company with interconnected plants, or even a whole economy forming a network of many production units.1 Any theory of production networks has to analyse the relation between the efficiency of the individual units (unit efficiency) and that of the whole production system (system efficiency) formed by their network. Economic literature as well as literature on efficiency measurement with methods of data envelopment analysis (DEA) commonly refer to Shephard (1970) as their production theoretical foundation. This theory, with axioms based on input or output possibility sets, has been further developed and applied mainly in the context of general or e.g. agrarian economics. It has not found much interest in business and engineering sciences, neither in the theory of business economics nor in research and teaching on production and operations management. This is in stark contrast to the alternative approach of Activity Analysis of Production and Allocation, documented in the proceedings of a conference edited by Tjalling Koopmans in 1951 (cf. Fandel (1991)). In this book, the foundations of linear programming as well as of efficiency analysis were laid by several authors, in particular by Dantzig (1951) and by Koopmans (1951) himself. To date, numerous mathematical models and methods based on this origin have been developed to deal with economic planning, scheduling, and accounting problems. The use of such models and methods is common practice in larger companies of industries that are heavily affected by coupled production or characterised by a network of interconnected plants, like the chemical or iron and steel industries (Dyckhoff and Souren 2023, p. 1043). Activity analysis is furthermore the standard approach to modelling production networks in the literature of sustainable production and supply chain management since the 1990s (Thies et al. 2021). Therefore, it should be ideally suited to form the starting point for a general network production theory that allows for efficiency analyses and can serve as a building block for network modelling. Koopmans (1951) developed activity analysis for polyhedral cone technologies, thus excluding general technologies which may be non-convex, even discrete. Until today, nonlinear generalisations of his activity analysis are notably rare, despite early 1 Andersson and Johanson (2018, p. 501) assert: “In recent decades, there has been a remarkable growth in the number of production units of firms such as IKEA, Walmart and Apple to name a few such global networking firms. Most of the analysis of these network firms has been modeled by logistics and other operations-research analysts (…) and to a limited extent by researchers in business administration schools. Very little has been done in economics.”. 811 Efficiency ofactivities inproduction networks ofarbitrary… contributions by Hildenbrand (1966) and Wittmann (1968). This is especially true for literature dedicated to nonlinear production networks.2 There seems to be also a lack regarding theoretical foundations of efficiency measurement in networks.3 Thus, to reduce this research gap, our paper aims at developing a generic foundation for efficiency analysis in arbitrary network systems. This general network production theory generalises (linear) activity analysis, incorporates well-known methods of performance measurement to assess both unit and system efficiency, and is characterised by following features in particular: • Analysed networks may possess arbitrary structures that are formed by production units whose technologies may be non-convex and even discrete. • This network production theory generalises Koopmans’ linear activity analysis by using similar underlying modelling features and fundamental assumptions. • The modelling approach is not only theoretically founded, but also practical for analysing complex networks of e.g. supply chains or closed-loop systems with recycling. Like linear activity analysis it can easily be extended to dynamic analyses by taking account of inventories. • Common methods of efficiency measurement known from network DEA can be integrated into the theory so that improvements are facilitated. The structure of the paper is as follows: Sect.2 provides a general framework for activity-analytic modelling of network systems and efficiency concepts. Koopmans’ original approach is categorised as special case. By making pretty mild technological assumptions, properties of such general network systems are analysed in Sect.3. Theorems for the relationship between unit and system efficiency are proven, revealing severe differences between loose and tied networks. Characteristic effects of nonlinear technologies on efficiency are shown. Section4 applies these findings to network DEA. It is exemplarily demonstrated how meaningful efficiency scores can be calculated for individual units and the overall system. Furthermore, the influence of disposability assumptions for intermediate products on efficiency measures is analysed. Key findings are summarised and discussed in Sect.5, revealing potential future research directions. All proofs are presented in an appendix. In this paper, ℝ𝜅 denotes the 𝜅 -dimensional Euclidean space and ℝ𝜅 + its nonnegative orthant. Let 0 be the vector with all components equal to 0. For vectors 2 An exception is Kohli’s (2005) treatment of technology structures. Although “related to the activity analysis literature, and in particular to the network technology approach” (p. 103), it strongly differs from our approach. And even if the search for literature on production theoretical foundations of networks is not limited to nonlinear technologies the results are rare. An explorative topic search in the Web of Science categories Economics and Operations Research / Management Science with search string (‘network’ OR ‘stage’ OR ‘division’ OR ‘tier’) AND (‘activity analy*’ OR ‘production theor*’) leads only to 27 results, from which the above mentioned paper of Kohli and at most two other papers of Fandel (2001) and Andersson and Johanson (2018) are somewhat related to the theoretical modelling of production networks. 3 Literature on the state-of-the-art of network DEA is provided at the beginning of Sect.4. 812 H.Dyckhoff, R.Souren 𝐚,𝐛∈ℝ𝜅 , inequality 𝐚≥𝐛 (𝐚≫𝐛 ) means ak ≥ bk ( ak>bk ) for all k=1, …,𝜅 , whereas 𝐚>𝐛 denotes 𝐚≥𝐛 , 𝐚≠𝐛 . 2 Basic definitions andassumptions This section develops the basics of a general network production theory. Section2.1 presents our approach to model network systems and their constituting production units (thus generalising that of Dyckhoff (1992)), followed by a subsection that shows how Koopman’s activity analysis is embedded as special case. The third subsection deals with efficiency of production in general. 2.1 The production system andits units The input/output graph of Fig.1 presents the structure of a network (adopted from Kao (2017), p. 194f) that is composed of three production units and seven object types. Example 2.1 In Fig.1, production units A, B, and C are depicted by squares, object types by circles. They are connected by arrows which state that types #1, #2, and #5 are supplied from outside (and #5 partly produced by unit B) whereas types #4, #6, and #7 are delivered outwards (and #4 partly used by unit C). Production of unit A uses types #1 and #2 as process inputs to produce objects of types #3 and #4. Unit B produces objects of type #5 from inputs #2 and #3. Unit C transforms inputs #4 and #5 into outputs #6 and #7. Each arrow represents a flow of quantities of the object type depicted by the circle it is connected with. Qualitative changes of object types exclusively take place by the transformation processes of squares. Fig. 1 Complex production network with three units and seven object types 813 Efficiency ofactivities inproduction networks ofarbitrary… The production possibility set (PPS) of a unit or a system may be known as ‘blue-print technology’ from its construction by engineers or may be obtained from observed data, e.g. by DEA methods. With 𝜌∈ℙ denoting the units of a network, the (stand-alone) PPS of unit 𝜌 is defined by its feasible activities: It is determined through its technology and may be restricted by individual constraints (that are not caused by the network itself). Vector z comprises all object types k∈𝕂={1, ⋯ ,𝜅} that are relevant for the description of each single unit as well as the whole network as production system. Positive elements of activity vectors depict net outputs, negative elements net inputs. However, a lot of elements may be zero (and thus neglected) for certain activities z𝜌∈P𝜌 of a specific production unit (e.g. k = 1, 2, 3 for unit 𝜌=C of Fig.1). Assumption 2.2 Each PPS P𝜌(𝜌∈ℙ) is non-empty and closed. Further fundamental properties will be assumed later on. In any case, to derive corresponding properties of the whole production system it is necessary to model the connections between the individual units existing in the network. Vector z𝕊∈ℝ𝜅 denotes total input and output quantities of the whole system. Assumption 2.3 For each object type k∈𝕂 the flows of process inputs and outputs z𝜌 of all individual production units 𝜌∈ℙ are balanced with the total system input and output z𝕊 within the considered production period such that differences result in a change Δsk of stock of this object type: The total quantity of any object type that occurs in the system must on the one hand stem from parts that are either procured from outside or produced inside or taken from stock and must on the other hand be used as parts that are consumed inside or delivered outwards or stored, alternatively. Immaterial objects like services cannot be stored so that Δsk=0 . Since this paper concentrates on static analysis, material objects taken from stock ( Δsk<0 ) can be subsumed under system inputs, whereas those stored ( Δsk>0 ) can be added to system outputs. Because it is supposed that no leakages or losses of object flows occur within the network, the activity of the whole system completely results from the activities of its single units. The PPS of the whole system is thus determined by: Example 2.4 With Δsk=0 , z=z𝕊 and z𝜌∈P𝜌 for 𝜌∈ℙ={A,B,C} ), Eq.(2) for the network of three units and seven object types in Fig.1 become: (1) P𝜌={ z ∈ ℝ 𝜅|Activity z can be realised by unit 𝜌} (2) Δ sk= ∑ 𝜌∈ ℙ z 𝜌 k−z 𝕊 k (3) P 𝕊= { z∈ℝ𝜅 | z= ∑ 𝜌 ∈ ℙ z𝜌 ,z𝜌∈P𝜌 ,𝜌∈ℙ } 814 H.Dyckhoff, R.Souren To avoid potential infeasibilities, it is common practice in the literature of economics and efficiency analysis to assume free disposability of inputs and outputs. E.g., in case of intermediate product #3 in Fig.1, one would assume zA 3 ≥−z B 3 . It is important to recognise, that free disposability is not postulated as a matter of principle in this paper. Instead, balances (2) and (3) explicitly record any potential surplus or shortfall of an object type as system output or input (or change in inventory, respectively). In cases that a surplus or shortfall of intermediate product #3 is not allowed, constraint z3 =z A 3 +z B 3 = 0 must be formulated as done above. Correspondingly, objects of a type that can only be supplied to the system but not delivered from it are restricted by zk ≤ 0 ; or by zk ≥ 0 in the opposite case. Regarding the example of Fig.1, flows of the three intermediate products are exogenously restricted so that the system PPS becomes: They are special cases of lower and upper bounds for system inputs or outputs: zk≤z k ≤zk ,that may exist in general but are often not binding. Definition 2.5 The network of a production system is loose when each unit of the system is free to choose its process inputs and outputs without any exogeneous constraints regarding the resulting input and output of the whole system. Otherwise it is called tied. In case (3), the PPS is a loose network that is uniquely determined by the combined object flows resulting from the unrestricted flows of its individual units. If it is tied as above one obtains: 2.2 Koopmans’ activity analysis asnetwork theory ofray technologies In extreme cases, the PPS of each production unit may be determined by a single basic activity a𝜌 = ( a 𝜌 1 ,…,a 𝜌 𝜅 ) ∈ℝ 𝜅 that can be arbitrarily multiplied such that: According to (3), the PPS of a corresponding loose network is then described by: z1 =z A 1 ,z 2 =z A 2 +z B 2 ,z A 3 +z B 3 =0, z 4 =z A 4 +z C 4 ,z 5 =z B 5 +z C 5 ,z 6 =z C 6 ,z 7 =z C 7 P 𝕊= { z∈ℝ7 | z= ∑ 𝜌∈ ℙ z𝜌,z𝜌∈P𝜌,𝜌∈ℙ={A,B,C},z3=0, z4≥0, z5≤0 } (4) P 𝕊= { z∈ℝ𝜅 | z= ∑ 𝜌 ∈ ℙ z𝜌 ,z𝜌∈P𝜌 ,𝜌∈ℙ,zk≤zk≤zk,k∈𝕂 } (5) P𝜌={ z ∈ℝ𝜅| z = a 𝜌 ⋅ 𝜆,𝜆 ≥ 0} (6) P 𝕊= { z∈ℝ𝜅 | z= ∑ 𝜌 ∈ ℙ a𝜌𝜆𝜌,𝜆𝜌≥0, 𝜌∈ℙ } 815 Efficiency ofactivities inproduction networks ofarbitrary… Such polyhedric cone technologies are the subject of Koopmans’ (1951) pioneering contribution. Hence, his Analysis of production as an efficient combination of activities can be interpreted as an analysis of networks of the specific type of ray technologies (5). Example 2.6 (cf. Müller-Merbach (1981), p. 66ff): Fig. 2 shows a one-stage network of five production units with four relevant object types. Each unit’s PPS (grey square) is determined as multiples of a corresponding basic activity (white square inside) with fixed input and output coefficients (placed at arrows nearby). According to PPS (6), flows of inputs and outputs of this network are completely described by following equations, with 𝜆𝜌≥0 for the activity levels of units 𝜌∈ℙ={A,B,C,D,E} : Network flows Production possibilities z1=z A 1+z B 1 z 2=zC 2+zD 2+z E 2 z A 3 +zC 3 +zD 3 =z 3 zA 4 +z B 4 +z D 4 +z E 4 =z 4 z A 1 =−𝜆A,zA 3 =𝜆A,zA 4 =2𝜆 A zB 1 =−𝜆 B ,z B 4 =5𝜆 B zC 2 =−𝜆C,z C 3 =2𝜆 C zD 2 =−𝜆 D ,z D 3 =𝜆 D ,z D 4 =3𝜆 D zE 2 =−𝜆 E ,z E 4 =7𝜆 E Koopmans (1951, p. 47ff) postulated four fundamental technological properties. They imply particular forms of polyhedral cone technologies that are reflected in respective properties of the matrices formed by elements a𝜌 k of the basic activities a𝜌 of production units 𝜌∈ℙ in (5) and (6). We pick up his first two postulates, and add only one weak, integrated version of the last two, though now all in terms of networks of general technologies.4 Postulates 2.7 Following properties should hold for any (loose or tied) production network and analogously for all its subsystems: Fig. 2 One-stage network of units with basic technologies 4 Postulates C and D of Koopmans (1951, pp. 53,55) differentiate two cases of networks without ‘intermediate commodities’ k and with pure ones ( zk=0 ), and in both cases a strong version (C1 and D1) from a weak one (C2 and D2). We only postulate the weak version and do not restrict our analysis to pure intermediate products. 816 H.Dyckhoff, R.Souren (a) Irreversibility of production: If z1, z 2∈P𝕊 such that z 1+ z 2=0 , then z1= z 2=0 , i.e. P𝕊∩−P𝕊⊆{0}. (b) Impossibility of the Land of Cockaigne: There exists no activity z∈P𝕊 with z>0 , i.e. P𝕊 ∩ℝ𝜅 + ⊆{0 } . (c) Possibility of production: Object types k=1, ⋯ ,𝜅 can be subdivided into three classes, namely primary factors, intermediate and final products. There exists an activity z∈P𝕊 with positive net output zk>0 for at least one type of final good so that P𝕊⧵ ℝ𝜅 − ≠ ∅ . With Koopmans (1951, p. 47) we do not claim “that in all uses of models of production these properties should be present. Rather, it is believed that in a broad class of cases it will be useful to employ models having these properties.” For example, it is obvious that a (hypothetical) network may violate Postulate 2.7a although each of its production units satisfies irreversibility. The same holds for Postulate 2.7b, which states ‘No output without input’ or ‘No free lunch’. A counterexample is given by three units A, B, and C with following activities: Remark 2.8 Postulate 2.7b prohibits that a free lunch would be achieved by combining several feasible activities. In a certain manner, the first two postulates reflect the Second Law of thermodynamics which implies that a perpetuum mobile is impossible. To be clear, this limitation is not an exogeneous restriction tying the network. In fact, it is an endogenous property that is inherent to all networks by naturally restricting each single PPS of all individual production units such that in total reversibility and free lunch cannot occur as long as labour or energy (precisely exergy) are relevant object types. 2.3 Efficiency ofproduction Before analysing properties of general networks, some fundamental aspects of production efficiency are considered next, for individual units as well as whole systems. It is required that preferences for production activities must be compatible with following partial preference order. Assumption 2.9 If z1 dominates z2 , production activity z1 is preferred to z2 , i.e. the first is better and the second worse than the other activity. The concrete meaning of this preference assumption depends on the specific definition of dominance. Here, dominance is defined by vector inequalities z1> z 2 , i.e. activity #1 has less input or more output than #2 for at least one object type, and not more input or less output else. Then, input and output are desirable objects (called z 𝕊=zA+zB+zC= ⎛ ⎜ ⎜ ⎝ −1 2 0 ⎞ ⎟ ⎟ ⎠ + ⎛ ⎜ ⎜ ⎝ 0 −1 2 ⎞ ⎟ ⎟ ⎠ + ⎛ ⎜ ⎜ ⎝ 2 0 −1 ⎞ ⎟ ⎟ ⎠ = ⎛ ⎜ ⎜ ⎝ 1 1 1 ⎞ ⎟ ⎟ ⎠ 823 Efficiency ofactivities inproduction networks ofarbitrary… 4 Efficiency measurement forpolyhedral networks Network production theory is now applied to networks whose units are characterised by (convex) polyhedric production technologies. This is a characteristic assumption of data envelopment analysis (DEA). It calculates the efficiency of a decisionmaking unit (DMU) by methods of Operations Research, in particular by linear programming. Respective research has been strongly growing within the past four decades (Panwar etal. 2022). DMUs consisting of a network of production units, called divisions, subunits, or components, are a main strand of DEA literature7 since the turn of the century (Liu etal. 2013, 2016; Lampe and Hilgers 2015), starting with pioneering contributions of Färe and Whittaker (1995) and Färe and Grosskopf (1996, 2000). Whereas so-called black box-DEA models ignore the internal structure of DMUs in measuring efficiency, network DEA models use the respective information to calculate not only efficiency scores for the individual units, but also a betterfounded score for the whole system (Kao 2017). General reviews of this literature are by Chen etal. (2013), Kao (2014), and Ratner etal. (2023), while Alves and Meza (2023) focus specifically on models with slacks-based efficiency measures. Let us assume that n DMUs j∈𝕁={1, …,n} with an identical structure of divisions 𝜌∈ℙ were observed and their input and output quantities a 𝜌 j= ( a𝜌 1j,…,a𝜌 𝜅j ) are known. Enveloping this data allows to construct an empirically determined PPS for each unit if certain properties of a minimum enveloping hull are assumed. In case of a linear hull one obtains: The corresponding PPS (4) of a tied network then becomes: Because each PPS (9) is closed, non-empty with 0∈P𝜌 , and convex with compact dominating sets, all respective propositions of Sect.3 are valid for (loose or tied) DEA networks in particular. As Sects.2 and 3 have shown, it is not difficult to model general production networks, including arbitrary ‘unstructured’ systems, and to define and analyse the efficiency of activities of such systems in principle. However, problems may arise when (9) P 𝜌= { z∈ℝ𝜅 | z= ∑ j∈𝕁 a𝜌 j𝜆𝜌 j,𝜆𝜌 j≥0, j∈𝕁 } ,𝜌∈ ℙ (10) P 𝕊= { z∈ℝ𝜅|z=∑ 𝜌∈ℙ z𝜌,z𝜌∈P𝜌,𝜌∈ℙ,zk≤zk≤zk,k∈𝕂 } = { z∈ℝ𝜅 | z= ∑ j∈𝕁 ∑ 𝜌∈ℙ a𝜌 j𝜆𝜌 j,𝜆𝜌 j≥0, j∈𝕁,𝜌∈ℙ,zk≤zk≤zk,k∈𝕂 } 7 Search for literature listed in the Web of Science—by combining terms ‘DEA’ or ‘data envelopment analy*’ with ‘network*’—results in more than 3000 papers up to the year 2023 most of which deal with the topic ‘network DEA’ (with about 400 even with this term in its title). 824 H.Dyckhoff, R.Souren trying to consistently define and calculate adequate efficiency measures e( z ) for the system as a whole as well as for each individual production unit. In his comprehensive book on network DEA, Kao (2017, p. 3) states in this regard: The whole-unit, or black-box, performance measurement is relatively simple to conduct, because only the inputs supplied to and the outputs produced by the DMU need to be considered, which makes a systematic expression of the model possible. The network system performance measurement, in contrast, is difficult to express using a general model, because different structures of the network production system are involved. (…) as a system becomes more complicated, the systematic expression of the model is only possible for certain specific types of system, and for other unstructured systems, this remains difficult. A main reason for this difficult problem seems to be that units of a network in general use different types of inputs and outputs each, in contrast to parallel systems (as in Fig.4)—so that outputs of one unit are inputs of others in particular. It is therefore not immediately obvious how to compare them in reducing different inputs or increasing different outputs. In any case, to define the “overall efficiency score” of a network (DMU) as arithmetic or harmonic mean of individual (divisional) scores, as proposed by Tone and Tsutsui (2009, p. 247), is inadequate for measuring the efficiency of a whole system as integrated entity (cf. Remark 3.8). Any mean is simply what it is: a numerical aggregate of efficiencies of possibly totally unconnected individual units, not necessarily belonging to the same or to any network. To address this problem, we first demonstrate how the efficiency of networks and their units can be measured reasonably. Then, the standard assumption of ‘free disposability’ is analysed for intermediate products regarding its influence on efficiency scores. Both subsections are based on a simple example and its modification that was used by Kao (2017, p. 178) for his overview of basic ideas in efficiency measurement for network systems. 4.1 New approach formeasuring network efficiency Example 4.1 Figure5 shows a two-stage tandem system—as “simplest structure of network systems” (Kao 2017, p. 3)—with two primary factors #1 and #2, two final products #3 and #4, and one pure intermediate product #5. Each of six DMUs is represented by a corresponding square within the two rectangles for the (divisions or) stages A and B (with the white squares representing basic activities and the grey rectangles the unit’s PPS). Note that activities of production units A and B are independent in principle. For example, a combination of basic activity A2 with that of B3 is feasible. The data of the six DMUs, all with this same structure, is assumed to be observed and is displayed in columns 2 to 6 of Table1 as well as at the respective input and 825 Efficiency ofactivities inproduction networks ofarbitrary… output arrows in Fig.5. Columns 7 to 11 of Table1 show efficiency scores for several DEA models which will be introduced and explained in the following. As common practice in DEA, index o∈𝕁 is used for the evaluated DMU and negative numbers are now avoided. With a A j = ( −xA 1j,−xA 2j, 0,0, yA 5j ) and a B j = ( 0,0, yB 3j,yB 4j,−xB 5j ) positive numbers denote quantities x 𝜌 ij and y 𝜌 rj of inputs i and outputs r for DMU j on stage ρ. To define adequate efficiency measures let us suppose that the DMUs have no influence on the demand for their final products. Thus, an efficiency measure for the DMU as a whole as well as for its production unit on the second stage seems appropriate which minimises their inputs, given the demand for final products. Hence, to coordinate production and consumption of the intermediate good, the first stage should also minimise its inputs, given the demand of the second stage for the intermediate good. Then, an adequate efficiency measure on stage B may be given by: e B o∶= e ( aB o ||| PB ) =min { 𝜃= −z5 xB 5o| z≥ ( 0,0, yB 3o,yB 4o,−xB 5o ) ,z∈PB } Fig. 5 Two-stage network with two tied units, five types of goods, and six DMUs Table 1 Data and efficiency of six DMUs with network structure of Fig.5 (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) DMU j aA 1j aA 2j aB 3j aB 4j aA 5j =−a B 5j eA j eB j e𝕆 j e𝕊 j e𝕊+ j 1– 1 – 2 2 3 3 0.75 0.72 1 0.54 0.54 2– 1 – 2 3 2 4 1 0.54 1 0.54 0.54 3– 2 – 1 2 3 4 1 0.54 1 0.54 0.54 4– 2 – 1 3 2 3 0.75 0.72 1 0.54 0.54 5– 2 – 4 5 5 3.6 0.45 1 1 0.45 0.45 6– 2 – 3 3 2 3 0.44 0.72 0.58 0.32 0.32 826 H.Dyckhoff, R.Souren By applying (9) we obtain following linear program: On stage A, the arithmetic mean of both input efficiencies is analogously determined by: Remark 4.2 With so-called “slacks” Δ x 𝜌 i =x 𝜌 io −x i (and Δy𝜌 r =yr−y𝜌 ro ) and because of such a type of minimisation program is identical to the so-called (input-oriented) additive “slack-based measure”, proposed by Tone (2001) and equivalently by Pastor etal. (1999). Efficiency scores of both stages and all six DMUs in Fig.5 are displayed in columns 7 and 8 of Table 1. They are similar to the radial efficiency scores of Kao (2017, p. 178). Only DMUs 2 and 3 are efficient on stage A and only DMU 5 on stage B. For #5 in Fig.5 being a pure intermediate product, network PPS (10) can be concretised to: e B o=𝑚𝑖𝑛𝜃 subject to 𝜃= x 5 xB 5o and ∑ j∈ 𝕁 xB 5j𝜆B j=x5≤xB 5 o ∑ j∈ 𝕁 yB rj𝜆B j=yr≥yB ro(r=3,4 ) 𝜆B j ≥0(j∈𝕁 ) e A o∶= e ( aA o ||| PA ) =𝑚𝑖𝑛𝜃 subject to 𝜃= 1 2 2 ∑ i=1 xi xA io and ∑ j∈ 𝕁 xA ij 𝜆A j=xi≤xA io(i=1,2 ) ∑ j∈ 𝕁 yA 5j𝜆A j=y5≥yA 5 o 𝜆A j ≥0(j∈𝕁 ) 1 2 2 ∑ i=1 xi xA io =1−1 2 ( ΔxA 1 xA 1o + ΔxA 2 xA 2o) 827 Efficiency ofactivities inproduction networks ofarbitrary… Since black-box DEA models assume identical activity levels 𝜆A j = 𝜆B j = 𝜆 j(j∈𝕁 ) for both stages (called ‘intensity variables’ in DEA literature), PPS (11) is reduced to: Vector z j∶= a A j +a B j = ( −x1j,−x2j,y3j,y4j,z5j ) represents columns 2 to 6 for each of the six rows of Table1. With z 5j=a A 5j +a B 5j = 0 the black-box model for DMU o∈𝕁 that is analogous to the models for stages A and B above is as follows: Column 9 of Table 1 displays the corresponding black-box efficiency scores, where all but DMU j=6 are efficient. This is because both production stages cannot exhaust all their individual production possibilities that exist if the linear hull of their data is valid. However, if one allows for independent combinations instead, i.e. 𝜆A j ≠𝜆 B j if necessary, even in case of pure intermediate product #5 according to PPS (11) the respective network DEA model becomes: (11) P 𝕊= { z∈ℝ5 | z= ∑ j∈𝕁 ( aA j𝜆A j+aB j𝜆B j ) ,𝜆A j,𝜆B j≥0, j∈𝕁,z5=0 } (12) P 𝕆= { z∈ℝ5|z=∑ j∈𝕁 ( aA j+aB j ) 𝜆j,𝜆j≥0, j∈𝕁,z5=0 } = { z∈ℝ5 | z= ∑ j∈𝕁 zj𝜆j,𝜆j≥0, j∈𝕁,z5=0 } (13) e 𝕆 o∶= e ( zo ||| P𝕆 ) =𝑚𝑖𝑛𝜃 subject to 𝜃= 1 2 2 ∑ i=1 xi x io ∑ j∈ 𝕁 xij𝜆j=xi≤xio (i=1,2 ) ∑ j∈ 𝕁 yrj𝜆j=yr≥yro (r=3,4 ) 𝜆j ≥ 0(j∈𝕁) (14) e 𝕊 o∶= e ( zo ||| P𝕊 ) =𝑚𝑖𝑛𝜃 subject to 𝜃= 1 2 2 ∑ i=1 xi xA io and ∑ j∈ 𝕁 xA ij 𝜆A j=xi≤xA io (i=1,2 ) 828 H.Dyckhoff, R.Souren Column 10 of Table1 displays the corresponding network efficiency scores. Now, none of the six DMUs acts efficiently (because they do not exhaust the production possibilities of their own units). Remark 4.3 This result contradicts common (black box) DEA knowledge which states that at least one DMU must be efficient (cf. e.g. Cooper etal. (2007)). It is crucial to notice that such a proposition is no longer valid for networks if every unit (or division) can freely choose from all activities that are possible for each DMU. Column 11 of Table1 shows the efficiency scores e𝕊+ j resulting from relaxing ties by z5≥0 , which allows for free disposability of the intermediate product. That does not change previous scores: e𝕊+ j =e𝕊 j. Remark 4.4 Network DEA models usually assume free disposability of excess output of intermediate products (e.g. Färe and Grosskopf 2000; Kao 2014; Lim and Zhu 2016); an exception is Tone and Tsutsui (2009, p. 246)). In case of Example 4.1 with a single intermediate product any excess output z5>0 can be reduced to zero ( z5=0 ) by decreasing certain activity levels of DMUs in stage A which does not lead to an increase of input of primary factors #1 and #2 whereas the production of stage B is not changed. Thus, there is always a solution without system output of the single intermediate product resulting in an identical efficiency score. 4.2 Influence offree disposability onefficiency scores This subsection investigates networks with more than one intermediate product. In DEA literature, intermediate products are often called links. Concluding their review of network DEA literature with further research directions Alves and Meza (2023, p. 2747) state: (…) it is suggested to analyze how the adoption of different types of links influences the potential conflicts between the processes resulting from the intermediate variables. That is, the second process may have to reduce its inputs (intermediate measures), in the case of an input orientation, to achieve an “efficient” status. Such an action, however, would imply a reduction in the outputs of the first stage, thus reducing the efficiency of this stage. In an output orientation, the opposite would occur. To expand the literature at this point, ∑ j∈ 𝕁 yB rj𝜆B j=yr≥yB ro (r=3,4 ) ∑ j∈ 𝕁 yA 5j𝜆A j− ∑ j∈ 𝕁 xB 5j𝜆B j=z5= 0 𝜆A j ,𝜆 B j ≥0(j∈𝕁 ) 829 Efficiency ofactivities inproduction networks ofarbitrary… the conflicting nature of intermediate measures and how the efficiency measure is affected could be studied. To study the influence of ties on intermediate products k∈𝕂 , two important cases are focused as before: (1) pure intermediate product ( zk=0 ) and (2) free disposability ( zk≥0 ). Example 4.5 Figure6 shows the simple two-stage network of Fig.5 (without indicating the observed basic activities of the six DMUs), now amended by a second intermediate product #6. Columns 2 to 6 of Table2 again contain the same data for the DMUs as before, but column 7 the new data for intermediate product #6. Here, instead of zk , we prefer to use nonnegative symbols for input xi and output yr . Columns 8 to 12 display the efficiency scores for the DEA models introduced in Sect.4.1, now applied to the network with two instead of one intermediate product. Obviously, the scores in column 10 of Table2 and column 9 of Table1 must be identical since the black-box DEA model (13) does not change because it ignores the interior network. Thus, except for DMU 6, all other five are still efficient in this respect. Fig. 6 Two-stage network of Fig.5, but with two intermediate products (instead of one) Table 2 Data and efficiency of six DMUs with network structure of Fig.6 (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) DMU jx A 1j xA 2j yB 3 j yB 4 j yA 5j =x B 5j yA 6j =x B 6j eA j eB j e𝕆 j e𝕊 j e𝕊+ j 1 1 2 2 3 3 3 0.75 0.81 1 0.69 0.68 2 1 2 3 2 4 4 1 0.61 1 0.61 0.61 3 2 1 2 3 4 3 1 0.72 1 0.97 0.89 4 2 1 3 2 3 2 0.75 1 1 0.71 0.71 5 2 4 5 5 3.6 4.5 0.56 1 1 0.58 0.56 6 2 3 3 2 3 5 0.73 0.63 0.58 0.36 0.36 830 H.Dyckhoff, R.Souren Both stage models as well as network model (14) require only slight changes regarding the second intermediate product #6. Stage A model now reads: DEA model for stage B becomes: Efficiency scores eA j and eB j of both stages are noted in columns 8 and 9 of Table2. Regarding stage A, solely the scores for DMUs 5 and 6 increase by a small amount, so that on stage A DMUs 2 and 3 are still efficient. With respect to stage B, DMU 4 is now also efficient (besides DMU 5 as before). The scores for the other four DMUs increase a bit. The network DEA model for the whole system with potential ties regarding both intermediate products reads: (15) e A o=𝑚𝑖𝑛𝜃 subject to 𝜃= 1 2 2 ∑ i=1 xi xA io and ∑ j∈ 𝕁 xA ij 𝜆A j=xi≤xA io (i=1,2 ) ∑ j∈ 𝕁 yA kj𝜆A j=yk≥yA ko (k=5,6 ) 𝜆A j ≥0(j∈𝕁 ) (16) e B o=𝑚𝑖𝑛𝜃 subject to 𝜃= 1 2 6 ∑ k=5 xk xB ko and ∑ j∈ 𝕁 xB kj𝜆B j=xk≤xB ko (k=5,6 ) ∑ j∈ 𝕁 yB rj𝜆B j=yr≥yB ro (r=3,4 ) 𝜆B j ≥0(j∈𝕁 ) (17) e 𝕊 o=𝑚𝑖𝑛𝜃 subject to 𝜃= 1 2 2 ∑ i=1 xi xA io and ∑ j∈ 𝕁 xA ij 𝜆A j=xi≤xA io (i=1,2 ) 831 Efficiency ofactivities inproduction networks ofarbitrary… Columns 11 and 12 of Table2 show (input) efficiencies e𝕊 j and e𝕊+ j of the DMUs for the two cases that both intermediate products are either pure ( zk=0 ) or else freely disposable ( zk≥0 ). As for the individual stages A and B, system scores are now larger with two instead of one intermediate product in Table1. Nonetheless, no DMU is efficient. Best scores are achieved for DMU 3. While in Table1 all efficiency scores of DMUs are smaller than each of the two stage scores, or at most equal, both system scores of DMU 3 in Table2 are now larger than the corresponding score of stage B, whereas for DMU 5 its system score is larger than the individual score of stage A solely in case of pure intermediate products. DMU 3 is also an example where its system efficiency in case of free disposability ( e𝕊+ j =0.89 ) is distinctly smaller than in case of pure intermediate products ( e𝕊 j =0.97 ) . Remark 4.7 It is important to notice that network DEA models known from literature often do not measure (strong Pareto–Koopmans) efficiency but merely a certain kind of weak or directional efficiency. Accordingly, by using input-oriented measures the examples of Sect.4 ignore possible improvements of individual outputs on stages A and B. In particular, multi-stage DMUs cannot get a better efficiency score by producing more of intermediate ‘goods’ than required by the subsequent stage. This implies that quantities of objects of a certain type of intermediate are desirable (‘good’) whereas other quantities of the same object type are of no value (‘free’ or ‘neutral’), i.e. their desirability is not fixed but depends on their produced or consumed quantity (cf. Dyckhoff (2023b) in this regard). 5 Discussion andconclusions Network DEA is a rapidly growing topic in productivity and efficiency measurement. Most of the more recent work involves applications of existing or slightly modified model approaches in various sectors such as banking, education, energy, environment, or transportation. They usually consider simple, well-structured networks and rely on the traditional assumptions of DEA, i.e. convex polyhedral technologies in combination with radial or slacks-based efficiency measures. Due to a variety of underlying definitions of efficiency and different modelling assumptions about the network, the results may seem contradictory and confusing, although being formally correct. There is a lack of approaches for a general theory of production networks ∑ j∈ 𝕁 yB rj𝜆B j=yr≥yB ro (r=3,4 ) ∑ j∈ 𝕁 yA kj𝜆A j− ∑ j∈ 𝕁 xB kj𝜆B j=yA k−xB k=zk,zk≤zk≤zk(k=5,6 ) 𝜆A j ,𝜆 B j ≥0(j∈𝕁 ) 832 H.Dyckhoff, R.Souren that can provide basic results and guidelines for the systematic construction of practical network models and appropriate efficiency measurement. Against this background, our paper presents a generalisation of Koopman’s linear activity analysis as generic approach to model arbitrary, possibly unstructured networks of production units, that may be working stations, plants, companies, or even whole countries. Each unit’s production possibility set has to satisfy rather weak conditions, only. They should be fulfilled by common assumptions in the economic and production management literature. In particular, non-convex and even discrete technologies are allowed in principle. Free disposability of inputs or outputs is not presupposed per se but may be explicitly modelled by excess quantities allowing to analyse the influence of this assumption on production properties such as efficiency and productivity. Although our presentation is restricted to static considerations the approach can easily be extended to dynamic investigations because network flow eqs. (2) are based on balances that may integrate inventories in case of material inputs and outputs.8 A main question is concerned with the relation between system efficiency and unit efficiencies. Theorems 3.3 and 3.6 show that answers to this question crucially depend on possible restrictions to system inputs or outputs that form ties to network flows. In case of loose networks, i.e. no active ties, each production unit can freely choose its input and output quantities also from outside the system if it makes sense to act optimally. Then, in order for loose networks to act efficiently all its production units have to be efficient, too. Regarding the opposite direction, however, there exist loose networks where the system (or DMU) as a whole is inefficient although all their units (or divisions) may act efficiently. On the contrary, a tied network may act efficiently although all its units’ activities are inefficient. A specific reason why it is difficult to define reasonable measures for the (in)efficiency of an arbitrary network is that its units in general do not use the same types of inputs and outputs, so that it is not clear how to compare them in reducing different inputs or increasing different outputs. Simple examples in Sect.4 prove that efficiency scores of a system need not inevitably be located between the minimum and maximum of the individual unit scores (what every kind of mean would do per definition). On the contrary, DMUs 1 and 6 in Tables1 and 2 demonstrate cases with system efficiency less than each of both unit efficiencies: e 𝕊 j<min { eA j,eB j } . The opposite case e 𝕊 j>max { eA j,eB j } is implicated by Example 3.5. Since ‘purity’ of intermediate products k ( zk=0 ) as constraint appears at least as strong as their free disposability ( zk≥0 ), optimisation programs like (14) always lead to solutions where the network with pure intermediate products displays efficiency scores not less than in case of free disposability, i.e. e𝕊 j ≥e 𝕊+ j . Regarding the question how the efficiency of a network activity may indeed be influenced by 8 An example of dynamic activity analysis that is applied to network-based production planning is presented by Fandel (2001).