Efficiency of activities in production networks of arbitrary structures and technologies

Dyckho , Ha ald; Sou en, Raine
A icle — Published Ve sion
E iciency o ac i i ies in p oduc ion ne wo ks o a bi a y
s uc u es and echnologies
Jou nal o Business Economics
P o ided in Coope a ion wi h:
Sp inge Na u e
Sugges ed Ci a ion: Dyckho , Ha ald; Sou en, Raine (2025) : E iciency o ac i i ies in p oduc ion
ne wo ks o a bi a y s uc u es and echnologies, Jou nal o Business Economics, ISSN 1861-8928,
Sp inge , Be lin, Heidelbe g, Vol. 95, Iss. 6, pp. 809-838,
h ps://doi.o g/10.1007/s11573-025-01228-9
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Jou nal o Business Economics (2025) 95:809–838
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ORIGINAL PAPER
E iciency o ac i i ies inp oduc ion ne wo ks o a bi a y
s uc u es and echnologies
Ha aldDyckho 1 · Raine Sou en2
Accep ed: 17 Feb ua y 2025 / Published online: 17 Ma ch 2025
© The Au ho (s) 2025
Abs ac
P oduc ion akes place in complex ne wo ks. An impo an ques ion is how he
e iciency o he whole ne wo k is ela ed o ha o he indi idual p oduc ion uni s
o ming he ne wo k. A gene al ne wo k p oduc ion heo y wi h ollowing cha ac e -
is ics is de eloped. Analysed ne wo ks may possess a bi a y s uc u es wi h uni s
whose echnologies may be non-con ex and e en disc e e. The heo y gene alises
Koopmans’ linea ac i i y analysis based on simila unde lying modelling ea u es
and undamen al assump ions. The modelling app oach is mo e sui able o analys-
ing ne wo ks han common ones. Me hods o e iciency measu emen known om
ne wo k da a en elopmen analysis a e in eg a ed in o he heo y. I is shown ha
calcula ing an o e all e iciency sco e o a ne wo k as a e age o indi idual sco es
o i s uni s is inapp op ia e. The ela ionship be ween he e iciency o a ne wo k
ac i i y and ha o subsys ems and uni s s ongly depends on he ex en o which he
indi idual p oduc ion uni s a e ee o choose hei inpu and ou pu quan i ies, i.e.
whe he he ne wo k is loose o ied. Especially in cases whe e lows o in e media e
p oduc s a e cons ained ins ead o eely disposable, he explici modelling o hei
o e p oduc ion helps o analyse hei in luence on e iciency sco es.
Keywo ds Ac i i y analysis· P oduc ion ne wo ks· E iciency analysis· Non-
con ex echnology· Ne wo k DEA
JEL Classi ica ion C14· C67· D24· L25
* Ha ald Dyckho
Dyc[email p o ec ed]h-aachen.de
Raine Sou en
Raine [email p o ec ed]
1 School o Business andEconomics (Fo me Chai o Business Theo y, Sus ainable
P oduc ion, andIndus ial Con ol), RWTH Aachen Technical Uni e si y, Temple g aben 64,
52056Aachen, Ge many
2 G oup o Sus ainable P oduc ion andLogis ics Managemen , Ilmenau Uni e si y o Technology,
Pos box100565, 98684Ilmenau, Ge many
810
H.Dyckho , R.Sou en
1 In oduc ion
P oduc ion is a human-di ec ed and con olled p ocess ha uses and ans o ms
selec ed objec s (including se ices) as inpu o c ea e new objec s ha eme ge as
ou pu om he p ocess. I is aimed a alue c ea ion, i.e., o al ad an ages gene -
a ed shall ou weigh o al disad an ages. Usually, he ans o ma ion akes place in
sys ems consis ing o se e al uni s ha p oduce la gely independen ly om each
o he , excep o being linked by objec s ha a e deli e ed as ou pu om one uni
and ecei ed as inpu by ano he uni wi hin he sys em. Such a sys em may be a
machine ac o y wi h wo king s a ions, a chemical company wi h in e connec ed
plan s, o e en a whole economy o ming a ne wo k o many p oduc ion uni s.1 Any
heo y o p oduc ion ne wo ks has o analyse he ela ion be ween he e iciency o
he indi idual uni s (uni e iciency) and ha o he whole p oduc ion sys em (sys em
e iciency) o med by hei ne wo k.
Economic li e a u e as well as li e a u e on e iciency measu emen wi h me hods
o da a en elopmen analysis (DEA) commonly e e o Shepha d (1970) as hei
p oduc ion heo e ical ounda ion. This heo y, wi h axioms based on inpu o ou -
pu possibili y se s, has been u he de eloped and applied mainly in he con ex o
gene al o e.g. ag a ian economics. I has no ound much in e es in business and
enginee ing sciences, nei he in he heo y o business economics no in esea ch
and eaching on p oduc ion and ope a ions managemen .
This is in s a k con as o he al e na i e app oach o Ac i i y Analysis o P o-
duc ion and Alloca ion, documen ed in he p oceedings o a con e ence edi ed by
Tjalling Koopmans in 1951 (c . Fandel (1991)). In his book, he ounda ions o
linea p og amming as well as o e iciency analysis we e laid by se e al au ho s,
in pa icula by Dan zig (1951) and by Koopmans (1951) himsel . To da e, nume -
ous ma hema ical models and me hods based on his o igin ha e been de eloped o
deal wi h economic planning, scheduling, and accoun ing p oblems. The use o such
models and me hods is common p ac ice in la ge companies o indus ies ha a e
hea ily a ec ed by coupled p oduc ion o cha ac e ised by a ne wo k o in e con-
nec ed plan s, like he chemical o i on and s eel indus ies (Dyckho and Sou en
2023, p. 1043). Ac i i y analysis is u he mo e he s anda d app oach o model-
ling p oduc ion ne wo ks in he li e a u e o sus ainable p oduc ion and supply chain
managemen since he 1990s (Thies e  al. 2021). The e o e, i should be ideally
sui ed o o m he s a ing poin o a gene al ne wo k p oduc ion heo y ha allows
o e iciency analyses and can se e as a building block o ne wo k modelling.
Koopmans (1951) de eloped ac i i y analysis o polyhed al cone echnologies,
hus excluding gene al echnologies which may be non-con ex, e en disc e e. Un il
oday, nonlinea gene alisa ions o his ac i i y analysis a e no ably a e, despi e ea ly
1 Ande sson and Johanson (2018, p. 501) asse : “In ecen decades, he e has been a ema kable g ow h
in he numbe o p oduc ion uni s o i ms such as IKEA, Walma and Apple o name a ew such global
ne wo king i ms. Mos o he analysis o hese ne wo k i ms has been modeled by logis ics and o he
ope a ions- esea ch analys s (…) and o a limi ed ex en by esea che s in business adminis a ion
schools. Ve y li le has been done in economics.”.
811
E iciency o ac i i ies inp oduc ion ne wo ks o a bi a y…
con ibu ions by Hildenb and (1966) and Wi mann (1968). This is especially ue
o li e a u e dedica ed o nonlinea p oduc ion ne wo ks.2 The e seems o be also a
lack ega ding heo e ical ounda ions o e iciency measu emen in ne wo ks.3
Thus, o educe his esea ch gap, ou pape aims a de eloping a gene ic ounda-
ion o e iciency analysis in a bi a y ne wo k sys ems. This gene al ne wo k p o-
duc ion heo y gene alises (linea ) ac i i y analysis, inco po a es well-known me h-
ods o pe o mance measu emen o assess bo h uni and sys em e iciency, and is
cha ac e ised by ollowing ea u es in pa icula :
• Analysed ne wo ks may possess a bi a y s uc u es ha a e o med by p oduc-
ion uni s whose echnologies may be non-con ex and e en disc e e.
• This ne wo k p oduc ion heo y gene alises Koopmans’ linea ac i i y analysis
by using simila unde lying modelling ea u es and undamen al assump ions.
• The modelling app oach is no only heo e ically ounded, bu also p ac ical o
analysing complex ne wo ks o e.g. supply chains o closed-loop sys ems wi h
ecycling. Like linea ac i i y analysis i can easily be ex ended o dynamic anal-
yses by aking accoun o in en o ies.
• Common me hods o e iciency measu emen known om ne wo k DEA can be
in eg a ed in o he heo y so ha imp o emen s a e acili a ed.
The s uc u e o he pape is as ollows: Sec .2 p o ides a gene al amewo k o
ac i i y-analy ic modelling o ne wo k sys ems and e iciency concep s. Koopmans’
o iginal app oach is ca ego ised as special case. By making p e y mild echnologi-
cal assump ions, p ope ies o such gene al ne wo k sys ems a e analysed in Sec .3.
Theo ems o he ela ionship be ween uni and sys em e iciency a e p o en, e eal-
ing se e e di e ences be ween loose and ied ne wo ks. Cha ac e is ic e ec s o
nonlinea echnologies on e iciency a e shown. Sec ion4 applies hese indings o
ne wo k DEA. I is exempla ily demons a ed how meaning ul e iciency sco es can
be calcula ed o indi idual uni s and he o e all sys em. Fu he mo e, he in luence
o disposabili y assump ions o in e media e p oduc s on e iciency measu es is
analysed. Key indings a e summa ised and discussed in Sec .5, e ealing po en ial
u u e esea ch di ec ions. All p oo s a e p esen ed in an appendix.
In his pape ,
ℝ𝜅
deno es he
𝜅
-dimensional Euclidean space and
ℝ𝜅
+
i s non-
nega i e o han . Le 0 be he ec o wi h all componen s equal o 0. Fo ec o s
2 An excep ion is Kohli’s (2005) ea men o echnology s uc u es. Al hough “ ela ed o he ac i i y
analysis li e a u e, and in pa icula o he ne wo k echnology app oach” (p. 103), i s ongly di e s om
ou app oach. And e en i he sea ch o li e a u e on p oduc ion heo e ical ounda ions o ne wo ks
is no limi ed o nonlinea echnologies he esul s a e a e. An explo a i e opic sea ch in he Web o
Science ca ego ies Economics and Ope a ions Resea ch / Managemen Science wi h sea ch s ing (‘ne -
wo k’ OR ‘s age’ OR ‘di ision’ OR ‘ ie ’) AND (‘ac i i y analy*’ OR ‘p oduc ion heo *’) leads only
o 27 esul s, om which he abo e men ioned pape o Kohli and a mos wo o he pape s o Fandel
(2001) and Ande sson and Johanson (2018) a e somewha ela ed o he heo e ical modelling o p oduc-
ion ne wo ks.
3 Li e a u e on he s a e-o - he-a o ne wo k DEA is p o ided a he beginning o Sec .4.
812
H.Dyckho , R.Sou en
𝐚,𝐛∈ℝ𝜅
, inequali y
𝐚≥𝐛
(𝐚≫𝐛
) means
ak
≥
bk
(
ak>bk
) o all
k=1, …,𝜅
,
whe eas
𝐚>𝐛
deno es
𝐚≥𝐛
,
𝐚≠𝐛
.
2 Basic de ini ions andassump ions
This sec ion de elops he basics o a gene al ne wo k p oduc ion heo y. Sec ion2.1
p esen s ou app oach o model ne wo k sys ems and hei cons i u ing p oduc ion
uni s ( hus gene alising ha o Dyckho (1992)), ollowed by a subsec ion ha
shows how Koopman’s ac i i y analysis is embedded as special case. The hi d sub-
sec ion deals wi h e iciency o p oduc ion in gene al.
2.1 The p oduc ion sys em andi s uni s
The inpu /ou pu g aph o Fig.1 p esen s he s uc u e o a ne wo k (adop ed om
Kao (2017), p. 194 ) ha is composed o h ee p oduc ion uni s and se en objec
ypes.
Example 2.1 In Fig.1, p oduc ion uni s A, B, and C a e depic ed by squa es, objec
ypes by ci cles. They a e connec ed by a ows which s a e ha ypes #1, #2, and #5
a e supplied om ou side (and #5 pa ly p oduced by uni B) whe eas ypes #4, #6,
and #7 a e deli e ed ou wa ds (and #4 pa ly used by uni C). P oduc ion o uni A
uses ypes #1 and #2 as p ocess inpu s o p oduce objec s o ypes #3 and #4. Uni B
p oduces objec s o ype #5 om inpu s #2 and #3. Uni C ans o ms inpu s #4 and
#5 in o ou pu s #6 and #7. Each a ow ep esen s a low o quan i ies o he objec
ype depic ed by he ci cle i is connec ed wi h. Quali a i e changes o objec ypes
exclusi ely ake place by he ans o ma ion p ocesses o squa es.
Fig. 1 Complex p oduc ion ne wo k wi h h ee uni s and se en objec ypes

813
E iciency o ac i i ies inp oduc ion ne wo ks o a bi a y…
The p oduc ion possibili y se (PPS) o a uni o a sys em may be known as
‘blue-p in echnology’ om i s cons uc ion by enginee s o may be ob ained om
obse ed da a, e.g. by DEA me hods. Wi h
𝜌∈ℙ
deno ing he uni s o a ne wo k,
he (s and-alone) PPS o uni
𝜌
is de ined by i s easible ac i i ies:
I is de e mined h ough i s echnology and may be es ic ed by indi idual con-
s ain s ( ha a e no caused by he ne wo k i sel ). Vec o
z
comp ises all objec
ypes
k∈𝕂={1,
⋯
,𝜅}
ha a e ele an o he desc ip ion o each single uni as
well as he whole ne wo k as p oduc ion sys em. Posi i e elemen s o ac i i y ec-
o s depic ne ou pu s, nega i e elemen s ne inpu s. Howe e , a lo o elemen s may
be ze o (and hus neglec ed) o ce ain ac i i ies
z𝜌∈P𝜌
o a speci ic p oduc ion
uni (e.g. k = 1, 2, 3 o uni
𝜌=C
o Fig.1).
Assump ion 2.2 Each PPS
P𝜌(𝜌∈ℙ)
is non-emp y and closed.
Fu he undamen al p ope ies will be assumed la e on. In any case, o de i e
co esponding p ope ies o he whole p oduc ion sys em i is necessa y o model
he connec ions be ween he indi idual uni s exis ing in he ne wo k. Vec o
z𝕊∈ℝ𝜅
deno es o al inpu and ou pu quan i ies o he whole sys em.
Assump ion 2.3 Fo each objec ype
k∈𝕂
he lows o p ocess inpu s and ou pu s
z𝜌
o all indi idual p oduc ion uni s
𝜌∈ℙ
a e balanced wi h he o al sys em inpu
and ou pu
z𝕊
wi hin he conside ed p oduc ion pe iod such ha di e ences esul in
a change
Δsk
o s ock o his objec ype:
The o al quan i y o any objec ype ha occu s in he sys em mus on he one
hand s em om pa s ha a e ei he p ocu ed om ou side o p oduced inside o
aken om s ock and mus on he o he hand be used as pa s ha a e consumed
inside o deli e ed ou wa ds o s o ed, al e na i ely. Imma e ial objec s like se ices
canno be s o ed so ha
Δsk=0
. Since his pape concen a es on s a ic analysis,
ma e ial objec s aken om s ock (
Δsk<0
) can be subsumed unde sys em inpu s,
whe eas hose s o ed (
Δsk>0
) can be added o sys em ou pu s. Because i is sup-
posed ha no leakages o losses o objec lows occu wi hin he ne wo k, he ac i -
i y o he whole sys em comple ely esul s om he ac i i ies o i s single uni s. The
PPS o he whole sys em is hus de e mined by:
Example 2.4 Wi h
Δsk=0
,
z=z𝕊
and
z𝜌∈P𝜌
o
𝜌∈ℙ={A,B,C}
), Eq.(2) o
he ne wo k o h ee uni s and se en objec ypes in Fig.1 become:
(1)
P𝜌={
z
∈
ℝ
𝜅|Ac i i y
z
can be ealised by uni 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.Dyckho , R.Sou en
To a oid po en ial in easibili ies, i is common p ac ice in he li e a u e o econom-
ics and e iciency analysis o assume ee disposabili y o inpu s and ou pu s. E.g., in
case o in e media e p oduc #3 in Fig.1, one would assume
zA
3
≥−z
B
3
. I is impo an
o ecognise, ha ee disposabili y is no pos ula ed as a ma e o p inciple in his
pape . Ins ead, balances (2) and (3) explici ly eco d any po en ial su plus o sho all
o an objec ype as sys em ou pu o inpu (o change in in en o y, espec i ely). In
cases ha a su plus o sho all o in e media e p oduc #3 is no allowed, cons ain
z3
=z
A
3
+z
B
3
=
0
mus be o mula ed as done abo e. Co espondingly, objec s o a
ype ha can only be supplied o he sys em bu no deli e ed om i a e es ic ed by
zk
≤
0
; o by
zk
≥
0
in he opposi e case. Rega ding he example o Fig.1, lows o he
h ee in e media e p oduc s a e exogenously es ic ed so ha he sys em PPS becomes:
They a e special cases o lowe and uppe bounds o sys em inpu s o ou pu s:
zk≤z
k
≤zk
, ha may exis in gene al bu a e o en no binding.
De ini ion 2.5 The ne wo k o a p oduc ion sys em is loose when each uni o he
sys em is ee o choose i s p ocess inpu s and ou pu s wi hou any exogeneous con-
s ain s ega ding he esul ing inpu and ou pu o he whole sys em. O he wise i is
called ied.
In case (3), he PPS is a loose ne wo k ha is uniquely de e mined by he combined
objec lows esul ing om he un es ic ed lows o i s indi idual uni s. I i is ied as
abo e one ob ains:
2.2 Koopmans’ ac i i y analysis asne wo k heo y o  ay echnologies
In ex eme cases, he PPS o each p oduc ion uni may be de e mined by a single basic
ac i i y
a𝜌
=
(
a
𝜌
1
,…,a
𝜌
𝜅
)
∈ℝ
𝜅
ha can be a bi a ily mul iplied such ha :
Acco ding o (3), he PPS o a co esponding loose ne wo k is hen desc ibed 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
E iciency o ac i i ies inp oduc ion ne wo ks o a bi a y…
Such polyhed ic cone echnologies a e he subjec o Koopmans’ (1951) pionee -
ing con ibu ion. Hence, his Analysis o p oduc ion as an e icien combina ion o
ac i i ies can be in e p e ed as an analysis o ne wo ks o he speci ic ype o ay
echnologies (5).
Example 2.6 (c . Mülle -Me bach (1981), p. 66 ): Fig. 2 shows a one-s age ne -
wo k o i e p oduc ion uni s wi h ou ele an objec ypes. Each uni ’s PPS (g ey
squa e) is de e mined as mul iples o a co esponding basic ac i i y (whi e squa e
inside) wi h ixed inpu and ou pu coe icien s (placed a a ows nea by).
Acco ding o PPS (6), lows o inpu s and ou pu s o his ne wo k a e com-
ple ely desc ibed by ollowing equa ions, wi h
𝜆𝜌≥0
o he ac i i y le els o uni s
𝜌∈ℙ={A,B,C,D,E}
:
Ne wo k lows P oduc ion possibili ies
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. 47 ) pos ula ed ou undamen al echnological p ope ies.
They imply pa icula o ms o polyhed al cone echnologies ha a e e lec ed in
espec i e p ope ies o he ma ices o med by elemen s
a𝜌
k
o he basic ac i i ies
a𝜌
o p oduc ion uni s
𝜌∈ℙ
in (5) and (6). We pick up his i s wo pos ula es, and
add only one weak, in eg a ed e sion o he las wo, hough now all in e ms o
ne wo ks o gene al echnologies.4
Pos ula es 2.7 Following p ope ies should hold o any (loose o ied) p oduc ion
ne wo k and analogously o all i s subsys ems:
Fig. 2 One-s age ne wo k o
uni s wi h basic echnologies
4 Pos ula es C and D o Koopmans (1951, pp. 53,55) di e en ia e wo cases o ne wo ks wi hou ‘in e -
media e commodi ies’
k
and wi h pu e ones (
zk=0
), and in bo h cases a s ong e sion (C1 and D1) om
a weak one (C2 and D2). We only pos ula e he weak e sion and do no es ic ou analysis o pu e in e -
media e p oduc s.
816
H.Dyckho , R.Sou en
(a) I e e sibili y o p oduc ion: I
z1,
z
2∈P𝕊
such ha z
1+
z
2=0
, hen
z1=
z
2=0
, i.e.
P𝕊∩−P𝕊⊆{0}.
(b) Impossibili y o he Land o Cockaigne: The e exis s no ac i i y
z∈P𝕊
wi h
z>0
, i.e.
P𝕊
∩ℝ𝜅
+
⊆{0
}
.
(c) Possibili y o p oduc ion: Objec ypes
k=1,
⋯
,𝜅
can be subdi ided in o h ee
classes, namely p ima y ac o s, in e media e and inal p oduc s. The e exis s
an ac i i y
z∈P𝕊
wi h posi i e ne ou pu
zk>0
o a leas one ype o inal
good so ha
P𝕊⧵
ℝ𝜅
−
≠
∅
.
Wi h Koopmans (1951, p. 47) we do no claim “ ha in all uses o models o p o-
duc ion hese p ope ies should be p esen . Ra he , i is belie ed ha in a b oad class
o cases i will be use ul o employ models ha ing hese p ope ies.” Fo example, i
is ob ious ha a (hypo he ical) ne wo k may iola e Pos ula e 2.7a al hough each o
i s p oduc ion uni s sa is ies i e e sibili y. The same holds o Pos ula e 2.7b, which
s a es ‘No ou pu wi hou inpu ’ o ‘No ee lunch’. A coun e example is gi en by
h ee uni s A, B, and C wi h ollowing ac i i ies:
Rema k 2.8 Pos ula e 2.7b p ohibi s ha a ee lunch would be achie ed by combin-
ing se e al easible ac i i ies. In a ce ain manne , he i s wo pos ula es e lec
he Second Law o he modynamics which implies ha a pe pe uum mobile is
impossible. To be clea , his limi a ion is no an exogeneous es ic ion ying he
ne wo k. In ac , i is an endogenous p ope y ha is inhe en o all ne wo ks by
na u ally es ic ing each single PPS o all indi idual p oduc ion uni s such ha in
o al e e sibili y and ee lunch canno occu as long as labou o ene gy (p ecisely
exe gy) a e ele an objec ypes.
2.3 E iciency o p oduc ion
Be o e analysing p ope ies o gene al ne wo ks, some undamen al aspec s o p o-
duc ion e iciency a e conside ed nex , o indi idual uni s as well as whole sys ems.
I is equi ed ha p e e ences o p oduc ion ac i i ies mus be compa ible wi h ol-
lowing pa ial p e e ence o de .
Assump ion 2.9 I
z1
domina es
z2
, p oduc ion ac i i y
z1
is p e e ed o
z2
, i.e. he
i s is be e and he second wo se han he o he ac i i y.
The conc e e meaning o his p e e ence assump ion depends on he speci ic de i-
ni ion o dominance. He e, dominance is de ined by ec o inequali ies
z1>
z
2
, i.e.
ac i i y #1 has less inpu o mo e ou pu han #2 o a leas one objec ype, and no
mo e inpu o less ou pu else. Then, inpu and ou pu a e desi able objec s (called
z
𝕊=zA+zB+zC=
⎛
⎜
⎜
⎝
−1
2
0
⎞
⎟
⎟
⎠
+
⎛
⎜
⎜
⎝
0
−1
2
⎞
⎟
⎟
⎠
+
⎛
⎜
⎜
⎝
2
0
−1
⎞
⎟
⎟
⎠
=
⎛
⎜
⎜
⎝
1
1
1
⎞
⎟
⎟
⎠
823
E iciency o ac i i ies inp oduc ion ne wo ks o a bi a y…
4 E iciency measu emen o polyhed al ne wo ks
Ne wo k p oduc ion heo y is now applied o ne wo ks whose uni s a e cha ac e ised
by (con ex) polyhed ic p oduc ion echnologies. This is a cha ac e is ic assump-
ion o da a en elopmen analysis (DEA). I calcula es he e iciency o a decision-
making uni (DMU) by me hods o Ope a ions Resea ch, in pa icula by linea
p og amming. Respec i e esea ch has been s ongly g owing wi hin he pas ou
decades (Panwa e al. 2022). DMUs consis ing o a ne wo k o p oduc ion uni s,
called di isions, subuni s, o componen s, a e a main s and o DEA li e a u e7 since
he u n o he cen u y (Liu e al. 2013, 2016; Lampe and Hilge s 2015), s a ing
wi h pionee ing con ibu ions o Fä e and Whi ake (1995) and Fä e and G osskop
(1996, 2000). Whe eas so-called black box-DEA models igno e he in e nal s uc u e
o DMUs in measu ing e iciency, ne wo k DEA models use he espec i e in o ma-
ion o calcula e no only e iciency sco es o he indi idual uni s, bu also a be e -
ounded sco e o he whole sys em (Kao 2017). Gene al e iews o his li e a u e
a e by Chen e al. (2013), Kao (2014), and Ra ne e al. (2023), while Al es and
Meza (2023) ocus speci ically on models wi h slacks-based e iciency measu es.
Le us assume ha
n
DMUs
j∈𝕁={1, …,n}
wi h an iden ical s uc u e o di i-
sions
𝜌∈ℙ
we e obse ed and hei inpu and ou pu quan i ies
a
𝜌
j=
(
a𝜌
1j,…,a𝜌
𝜅j
)
a e known. En eloping his da a allows o cons uc an empi ically de e mined PPS
o each uni i ce ain p ope ies o a minimum en eloping hull a e assumed. In
case o a linea hull one ob ains:
The co esponding PPS (4) o a ied ne wo k hen becomes:
Because each PPS (9) is closed, non-emp y wi h
0∈P𝜌
, and con ex wi h com-
pac domina ing se s, all espec i e p oposi ions o Sec .3 a e alid o (loose o
ied) DEA ne wo ks in pa icula .
As Sec s.2 and 3 ha e shown, i is no di icul o model gene al p oduc ion ne -
wo ks, including a bi a y ‘uns uc u ed’ sys ems, and o de ine and analyse he e i-
ciency o ac i i ies o such sys ems in p inciple. Howe e , p oblems may a ise 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 Sea ch o li e a u e lis ed in he Web o Science—by combining e ms ‘DEA’ o ‘da a en elopmen
analy*’ wi h ‘ne wo k*’— esul s in mo e han 3000 pape s up o he yea 2023 mos o which deal wi h
he opic ‘ne wo k DEA’ (wi h abou 400 e en wi h his e m in i s i le).

824
H.Dyckho , R.Sou en
ying o consis en ly de ine and calcula e adequa e e iciency measu es
e(
z
)
o he
sys em as a whole as well as o each indi idual p oduc ion uni . In his comp ehen-
si e book on ne wo k DEA, Kao (2017, p. 3) s a es in his ega d:
The whole-uni , o black-box, pe o mance measu emen is ela i ely simple
o conduc , because only he inpu s supplied o and he ou pu s p oduced by
he DMU need o be conside ed, which makes a sys ema ic exp ession o he
model possible. The ne wo k sys em pe o mance measu emen , in con as ,
is di icul o exp ess using a gene al model, because di e en s uc u es o
he ne wo k p oduc ion sys em a e in ol ed. (…) as a sys em becomes mo e
complica ed, he sys ema ic exp ession o he model is only possible o ce -
ain speci ic ypes o sys em, and o o he uns uc u ed sys ems, his emains
di icul .
A main eason o his di icul p oblem seems o be ha uni s o a ne wo k in
gene al use di e en ypes o inpu s and ou pu s each, in con as o pa allel sys ems
(as in Fig.4)—so ha ou pu s o one uni a e inpu s o o he s in pa icula . I is
he e o e no immedia ely ob ious how o compa e hem in educing di e en inpu s
o inc easing di e en ou pu s.
In any case, o de ine he “o e all e iciency sco e” o a ne wo k (DMU) as a i h-
me ic o ha monic mean o indi idual (di isional) sco es, as p oposed by Tone and
Tsu sui (2009, p. 247), is inadequa e o measu ing he e iciency o a whole sys em
as in eg a ed en i y (c . Rema k 3.8). Any mean is simply wha i is: a nume ical
agg ega e o e iciencies o possibly o ally unconnec ed indi idual uni s, no neces-
sa ily belonging o he same o o any ne wo k.
To add ess his p oblem, we i s demons a e how he e iciency o ne wo ks and
hei uni s can be measu ed easonably. Then, he s anda d assump ion o ‘ ee dis-
posabili y’ is analysed o in e media e p oduc s ega ding i s in luence on e iciency
sco es. Bo h subsec ions a e based on a simple example and i s modi ica ion ha
was used by Kao (2017, p. 178) o his o e iew o basic ideas in e iciency meas-
u emen o ne wo k sys ems.
4.1 New app oach o measu ing ne wo k e iciency
Example 4.1 Figu e5 shows a wo-s age andem sys em—as “simples s uc u e o
ne wo k sys ems” (Kao 2017, p. 3)—wi h wo p ima y ac o s #1 and #2, wo inal
p oduc s #3 and #4, and one pu e in e media e p oduc #5. Each o six DMUs is
ep esen ed by a co esponding squa e wi hin he wo ec angles o he (di isions
o ) s ages A and B (wi h he whi e squa es ep esen ing basic ac i i ies and he g ey
ec angles he uni ’s PPS). No e ha ac i i ies o p oduc ion uni s A and B a e inde-
penden in p inciple. Fo example, a combina ion o basic ac i i y A2 wi h ha o
B3 is easible.
The da a o he six DMUs, all wi h his same s uc u e, is assumed o be obse ed
and is displayed in columns 2 o 6 o Table1 as well as a he espec i e inpu and
825
E iciency o ac i i ies inp oduc ion ne wo ks o a bi a y…
ou pu a ows in Fig.5. Columns 7 o 11 o Table1 show e iciency sco es o se -
e al DEA models which will be in oduced and explained in he ollowing.
As common p ac ice in DEA, index
o∈𝕁
is used o he e alua ed DMU and
nega i e numbe s a e now a oided. Wi h
a
A
j
=
(
−xA
1j,−xA
2j, 0,0, yA
5j
)
and
a
B
j
=
(
0,0, yB
3j,yB
4j,−xB
5j
)
posi i e numbe s deno e quan i ies x
𝜌
ij
and y
𝜌
j
o inpu s i and
ou pu s o DMU j on s age ρ.
To de ine adequa e e iciency measu es le us suppose ha he DMUs ha e no in lu-
ence on he demand o hei inal p oduc s. Thus, an e iciency measu e o he DMU
as a whole as well as o i s p oduc ion uni on he second s age seems app op ia e
which minimises hei inpu s, gi en he demand o inal p oduc s. Hence, o coo di-
na e p oduc ion and consump ion o he in e media e good, he i s s age should also
minimise i s inpu s, gi en he demand o he second s age o he in e media e good.
Then, an adequa e e iciency measu e on s age B may be gi en 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-s age ne wo k wi h wo ied uni s, i e ypes o goods, and six DMUs
Table 1 Da a and e iciency o six DMUs wi h ne wo k s uc u e o 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.Dyckho , R.Sou en
By applying (9) we ob ain ollowing linea p og am:
On s age A, he a i hme ic mean o bo h inpu e iciencies is analogously de e -
mined by:
Rema k 4.2 Wi h so-called “slacks”
Δ
x
𝜌
i
=x
𝜌
io
−x
i
(and
Δy𝜌
=y −y𝜌
o
) and because
o
such a ype o minimisa ion p og am is iden ical o he so-called (inpu -o ien ed)
addi i e “slack-based measu e”, p oposed by Tone (2001) and equi alen ly by Pas-
o e al. (1999).
E iciency sco es o bo h s ages and all six DMUs in Fig.5 a e displayed in col-
umns 7 and 8 o Table 1. They a e simila o he adial e iciency sco es o Kao
(2017, p. 178). Only DMUs 2 and 3 a e e icien on s age A and only DMU 5 on
s age B.
Fo #5 in Fig.5 being a pu e in e media e p oduc , ne wo k PPS (10) can be con-
c e ised o:
e
B
o=𝑚𝑖𝑛𝜃 subjec o 𝜃=
x
5
xB
5o
and
∑
j∈
𝕁
xB
5j𝜆B
j=x5≤xB
5
o
∑
j∈
𝕁
yB
j𝜆B
j=y ≥yB
o( =3,4
)
𝜆B
j
≥0(j∈𝕁
)
e
A
o∶= e
(
aA
o
|||
PA
)
=𝑚𝑖𝑛𝜃 subjec o 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
E iciency o ac i i ies inp oduc ion ne wo ks o a bi a y…
Since black-box DEA models assume iden ical ac i i y le els
𝜆A
j
=
𝜆B
j
=
𝜆
j(j∈𝕁
)
o bo h s ages (called ‘in ensi y a iables’ in DEA li e a u e), PPS (11) is educed
o:
Vec o
z
j∶= a
A
j
+a
B
j
=
(
−x1j,−x2j,y3j,y4j,z5j
)
ep esen s columns 2 o 6 o each
o he six ows o Table1. Wi h
z
5j=a
A
5j
+a
B
5j
=
0
he black-box model o DMU
o∈𝕁
ha is analogous o he models o s ages A and B abo e is as ollows:
Column 9 o Table 1 displays he co esponding black-box e iciency sco es,
whe e all bu DMU
j=6
a e e icien . This is because bo h p oduc ion s ages can-
no exhaus all hei indi idual p oduc ion possibili ies ha exis i he linea hull o
hei da a is alid. Howe e , i one allows o independen combina ions ins ead, i.e.
𝜆A
j
≠𝜆
B
j
i necessa y, e en in case o pu e in e media e p oduc #5 acco ding o PPS
(11) he espec i e ne wo k 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 subjec o 𝜃=
1
2
2
∑
i=1
xi
x
io
∑
j∈
𝕁
xij𝜆j=xi≤xio (i=1,2
)
∑
j∈
𝕁
y j𝜆j=y ≥y o ( =3,4
)
𝜆j
≥
0(j∈𝕁)
(14)
e
𝕊
o∶= e
(
zo
|||
P𝕊
)
=𝑚𝑖𝑛𝜃 subjec o 𝜃=
1
2
2
∑
i=1
xi
xA
io
and
∑
j∈
𝕁
xA
ij 𝜆A
j=xi≤xA
io (i=1,2
)
828
H.Dyckho , R.Sou en
Column 10 o Table1 displays he co esponding ne wo k e iciency sco es. Now,
none o he six DMUs ac s e icien ly (because hey do no exhaus he p oduc ion
possibili ies o hei own uni s).
Rema k 4.3 This esul con adic s common (black box) DEA knowledge which
s a es ha a leas one DMU mus be e icien (c . e.g. Coope e al. (2007)). I is
c ucial o no ice ha such a p oposi ion is no longe alid o ne wo ks i e e y uni
(o di ision) can eely choose om all ac i i ies ha a e possible o each DMU.
Column 11 o Table1 shows he e iciency sco es
e𝕊+
j esul ing om elaxing ies
by
z5≥0
, which allows o ee disposabili y o he in e media e p oduc . Tha does
no change p e ious sco es:
e𝕊+
j
=e𝕊
j.
Rema k 4.4 Ne wo k DEA models usually assume ee disposabili y o excess ou -
pu o in e media e p oduc s (e.g. Fä e and G osskop 2000; Kao 2014; Lim and
Zhu 2016); an excep ion is Tone and Tsu sui (2009, p. 246)). In case o Example 4.1
wi h a single in e media e p oduc any excess ou pu
z5>0
can be educed o ze o
(
z5=0
) by dec easing ce ain ac i i y le els o DMUs in s age A which does no
lead o an inc ease o inpu o p ima y ac o s #1 and #2 whe eas he p oduc ion o
s age B is no changed. Thus, he e is always a solu ion wi hou sys em ou pu o he
single in e media e p oduc esul ing in an iden ical e iciency sco e.
4.2 In luence o  ee disposabili y one iciency sco es
This subsec ion in es iga es ne wo ks wi h mo e han one in e media e p oduc . In
DEA li e a u e, in e media e p oduc s a e o en called links. Concluding hei e iew
o ne wo k DEA li e a u e wi h u he esea ch di ec ions Al es and Meza (2023, p.
2747) s a e:
(…) i is sugges ed o analyze how he adop ion o di e en ypes o links
in luences he po en ial con lic s be ween he p ocesses esul ing om he
in e media e a iables. Tha is, he second p ocess may ha e o educe i s
inpu s (in e media e measu es), in he case o an inpu o ien a ion, o achie e
an “e icien ” s a us. Such an ac ion, howe e , would imply a educ ion in he
ou pu s o he i s s age, hus educing he e iciency o his s age. In an ou pu
o ien a ion, he opposi e would occu . To expand he li e a u e a his poin ,
∑
j∈
𝕁
yB
j𝜆B
j=y ≥yB
o ( =3,4
)
∑
j∈
𝕁
yA
5j𝜆A
j−
∑
j∈
𝕁
xB
5j𝜆B
j=z5=
0
𝜆A
j
,𝜆
B
j
≥0(j∈𝕁
)

829
E iciency o ac i i ies inp oduc ion ne wo ks o a bi a y…
he con lic ing na u e o in e media e measu es and how he e iciency meas-
u e is a ec ed could be s udied.
To s udy he in luence o ies on in e media e p oduc s
k∈𝕂
, wo impo an
cases a e ocused as be o e: (1) pu e in e media e p oduc (
zk=0
) and (2) ee dis-
posabili y (
zk≥0
).
Example 4.5 Figu e6 shows he simple wo-s age ne wo k o Fig.5 (wi hou indi-
ca ing he obse ed basic ac i i ies o he six DMUs), now amended by a second
in e media e p oduc #6. Columns 2 o 6 o Table2 again con ain he same da a
o he DMUs as be o e, bu column 7 he new da a o in e media e p oduc #6.
He e, ins ead o
zk
, we p e e o use nonnega i e symbols o inpu
xi
and ou pu
y
. Columns 8 o 12 display he e iciency sco es o he DEA models in oduced in
Sec .4.1, now applied o he ne wo k wi h wo ins ead o one in e media e p oduc .
Ob iously, he sco es in column 10 o Table2 and column 9 o Table1 mus be
iden ical since he black-box DEA model (13) does no change because i igno es
he in e io ne wo k. Thus, excep o DMU 6, all o he i e a e s ill e icien in
his espec .
Fig. 6 Two-s age ne wo k o Fig.5, bu wi h wo in e media e p oduc s (ins ead o one)
Table 2 Da a and e iciency o six DMUs wi h ne wo k s uc u e o 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.Dyckho , R.Sou en
Bo h s age models as well as ne wo k model (14) equi e only sligh changes
ega ding he second in e media e p oduc #6. S age A model now eads:
DEA model o s age B becomes:
E iciency sco es
eA
j
and
eB
j
o bo h s ages a e no ed in columns 8 and 9 o Table2.
Rega ding s age A, solely he sco es o DMUs 5 and 6 inc ease by a small amoun ,
so ha on s age A DMUs 2 and 3 a e s ill e icien . Wi h espec o s age B, DMU 4
is now also e icien (besides DMU 5 as be o e). The sco es o he o he ou DMUs
inc ease a bi .
The ne wo k DEA model o he whole sys em wi h po en ial ies ega ding bo h
in e media e p oduc s eads:
(15)
e
A
o=𝑚𝑖𝑛𝜃 subjec o 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 subjec o 𝜃=
1
2
6
∑
k=5
xk
xB
ko
and
∑
j∈
𝕁
xB
kj𝜆B
j=xk≤xB
ko (k=5,6
)
∑
j∈
𝕁
yB
j𝜆B
j=y ≥yB
o ( =3,4
)
𝜆B
j
≥0(j∈𝕁
)
(17)
e
𝕊
o=𝑚𝑖𝑛𝜃 subjec o 𝜃=
1
2
2
∑
i=1
xi
xA
io
and
∑
j∈
𝕁
xA
ij 𝜆A
j=xi≤xA
io (i=1,2
)
831
E iciency o ac i i ies inp oduc ion ne wo ks o a bi a y…
Columns 11 and 12 o Table2 show (inpu ) e iciencies
e𝕊
j and
e𝕊+
j o he DMUs
o he wo cases ha bo h in e media e p oduc s a e ei he pu e (
zk=0
) o else
eely disposable (
zk≥0
). As o he indi idual s ages A and B, sys em sco es a e
now la ge wi h wo ins ead o one in e media e p oduc in Table1. None heless, no
DMU is e icien . Bes sco es a e achie ed o DMU 3. While in Table1 all e i-
ciency sco es o DMUs a e smalle han each o he wo s age sco es, o a mos
equal, bo h sys em sco es o DMU 3 in Table2 a e now la ge han he co espond-
ing sco e o s age B, whe eas o DMU 5 i s sys em sco e is la ge han he indi id-
ual sco e o s age A solely in case o pu e in e media e p oduc s. DMU 3 is also an
example whe e i s sys em e iciency in case o ee disposabili y (
e𝕊+
j
=0.89
)
is dis-
inc ly smalle han in case o pu e in e media e p oduc s (
e𝕊
j
=0.97
)
.
Rema k 4.7 I is impo an o no ice ha ne wo k DEA models known om li e a-
u e o en do no measu e (s ong Pa e o–Koopmans) e iciency bu me ely a ce ain
kind o weak o di ec ional e iciency. Acco dingly, by using inpu -o ien ed meas-
u es he examples o Sec .4 igno e possible imp o emen s o indi idual ou pu s on
s ages A and B. In pa icula , mul i-s age DMUs canno ge a be e e iciency sco e
by p oducing mo e o in e media e ‘goods’ han equi ed by he subsequen s age.
This implies ha quan i ies o objec s o a ce ain ype o in e media e a e desi able
(‘good’) whe eas o he quan i ies o he same objec ype a e o no alue (‘ ee’ o
‘neu al’), i.e. hei desi abili y is no ixed bu depends on hei p oduced o con-
sumed quan i y (c . Dyckho (2023b) in his ega d).
5 Discussion andconclusions
Ne wo k DEA is a apidly g owing opic in p oduc i i y and e iciency measu emen .
Mos o he mo e ecen wo k in ol es applica ions o exis ing o sligh ly modi ied
model app oaches in a ious sec o s such as banking, educa ion, ene gy, en i on-
men , o anspo a ion. They usually conside simple, well-s uc u ed ne wo ks and
ely on he adi ional assump ions o DEA, i.e. con ex polyhed al echnologies in
combina ion wi h adial o slacks-based e iciency measu es. Due o a a ie y o
unde lying de ini ions o e iciency and di e en modelling assump ions abou he
ne wo k, he esul s may seem con adic o y and con using, al hough being o mally
co ec . The e is a lack o app oaches o a gene al heo y o p oduc ion ne wo ks
∑
j∈
𝕁
yB
j𝜆B
j=y ≥yB
o ( =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.Dyckho , R.Sou en
ha can p o ide basic esul s and guidelines o he sys ema ic cons uc ion o p ac-
ical ne wo k models and app op ia e e iciency measu emen .
Agains his backg ound, ou pape p esen s a gene alisa ion o Koopman’s linea
ac i i y analysis as gene ic app oach o model a bi a y, possibly uns uc u ed ne -
wo ks o p oduc ion uni s, ha may be wo king s a ions, plan s, companies, o e en
whole coun ies. Each uni ’s p oduc ion possibili y se has o sa is y a he weak
condi ions, only. They should be ul illed by common assump ions in he economic
and p oduc ion managemen li e a u e. In pa icula , non-con ex and e en disc e e
echnologies a e allowed in p inciple. F ee disposabili y o inpu s o ou pu s is no
p esupposed pe se bu may be explici ly modelled by excess quan i ies allowing o
analyse he in luence o his assump ion on p oduc ion p ope ies such as e iciency
and p oduc i i y. Al hough ou p esen a ion is es ic ed o s a ic conside a ions he
app oach can easily be ex ended o dynamic in es iga ions because ne wo k low
eqs. (2) a e based on balances ha may in eg a e in en o ies in case o ma e ial
inpu s and ou pu s.8
A main ques ion is conce ned wi h he ela ion be ween sys em e iciency and
uni e iciencies. Theo ems 3.3 and 3.6 show ha answe s o his ques ion c ucially
depend on possible es ic ions o sys em inpu s o ou pu s ha o m ies o ne wo k
lows. In case o loose ne wo ks, i.e. no ac i e ies, each p oduc ion uni can eely
choose i s inpu and ou pu quan i ies also om ou side he sys em i i makes sense
o ac op imally. Then, in o de o loose ne wo ks o ac e icien ly all i s p oduc ion
uni s ha e o be e icien , oo. Rega ding he opposi e di ec ion, howe e , he e exis
loose ne wo ks whe e he sys em (o DMU) as a whole is ine icien al hough all
hei uni s (o di isions) may ac e icien ly. On he con a y, a ied ne wo k may ac
e icien ly al hough all i s uni s’ ac i i ies a e ine icien .
A speci ic eason why i is di icul o de ine easonable measu es o he (in)e i-
ciency o an a bi a y ne wo k is ha i s uni s in gene al do no use he same ypes o
inpu s and ou pu s, so ha i is no clea how o compa e hem in educing di e en
inpu s o inc easing di e en ou pu s. Simple examples in Sec .4 p o e ha e i-
ciency sco es o a sys em need no ine i ably be loca ed be ween he minimum and
maximum o he indi idual uni sco es (wha e e y kind o mean would do pe de i-
ni ion). On he con a y, DMUs 1 and 6 in Tables1 and 2 demons a e cases wi h
sys em e iciency less han each o bo h uni e iciencies:
e
𝕊
j<min
{
eA
j,eB
j
}
. The
opposi e case
e
𝕊
j>max
{
eA
j,eB
j
}
is implica ed by Example 3.5.
Since ‘pu i y’ o in e media e p oduc s
k
(
zk=0
) as cons ain appea s a leas as
s ong as hei ee disposabili y (
zk≥0
), op imisa ion p og ams like (14) always
lead o solu ions whe e he ne wo k wi h pu e in e media e p oduc s displays e i-
ciency sco es no less han in case o ee disposabili y, i.e.
e𝕊
j
≥e
𝕊+
j
. Rega ding he
ques ion how he e iciency o a ne wo k ac i i y may indeed be in luenced by
8 An example o dynamic ac i i y analysis ha is applied o ne wo k-based p oduc ion planning is p e-
sen ed by Fandel (2001).