computing fuzzy process efficiency in parallel systems
TRANSCRIPT
Fuzzy Optim Decis MakingDOI 10.1007/s10700-013-9170-0
Computing fuzzy process efficiency in parallel systems
Sebastián Lozano
© Springer Science+Business Media New York 2013
Abstract This paper deals with parallel process systems in which the input and out-put data are fuzzy. The α-level based approach is used to compute the fuzzy systemefficiency and a simple procedure is proposed to estimate the fuzzy efficiency of thedifferent processes. The main contribution of the paper is estimating the latter takinginto account the variability of the process efficiencies compatible with a given value ofthe system efficiency. This variability comes from the existence of alternative optimalweights in the system efficiency multiplier network DEA models. The computation ofthe fuzzy system efficiency involves one Linear and one Non-linear Program for eachα-cut while the computation of each process efficiency requires solving just a coupleof related Linear Programs for each α-cut. The proposed approach is illustrated witha parallel systems dataset extracted from the literature.
Keywords Network DEA · Fuzzy data · Parallel processes · Process efficiency
1 Introduction
Data Envelopment Analysis (DEA) is a non-parametric technique for assessing therelative efficiency of a number of comparable entities generally termed Decision Mak-ing Units (DMUs). There are a number of DEA approaches that consider that each
S. Lozano (B)Department of Industrial Management, University of Seville, Seville, Spaine-mail: [email protected]
S. LozanoEscuela Superior de Ingenieros, Camino de los Descubrimientos, s/n, 41092 Sevilla, Spain
123
S. Lozano
DMU is not a single-process black box but consists, instead, of different interrelatedprocesses. The main advantage of this type of network DEA approaches is that theyallow a more fine-grained analysis than conventional single-process DEA.
The literature on Network DEA has increased significantly in the last decade. Sem-inal papers (e.g. Färe and Grosskopf 2000) were followed by a number of papersmainly dealing with two-stage systems (e.g. Kao and Hwang 2008), parallel processsystems (e.g. Kao 2012), general multistage processes (e.g. Kao and Hwang 2010),network SBM (e.g. Tone and Tsutsui 2009), network DEA with undesirable outputs(e.g. Lozano et al. 2013), network cost efficiency (e.g. Lozano 2011), etc.
A research topic within network DEA that has not yet received too much attentionso far is network DEA with fuzzy data. There are a good number of fuzzy approachesto single-process DEA (see Hatami-Marbini et al. 2011 for a review) but only a fewnetwork DEA approaches that consider imprecise or fuzzy data. Thus, Kao and Liu(2011) study two-stage systems with fuzzy data, Kao and Lin (2012) study parallelprocess systems with fuzzy data and Ashrafi and Jaafar (2011) consider interval dataand propose models for two-stage and parallel-process systems. In this paper, theα-level based approach in Kao and Lin (2012) is reviewed and a new method ofcomputing the process efficiencies is proposed.
The structure of the paper is the following. In Sect. 2 the required notation isintroduced. In Sect. 3, the Kao and Lin (2012) fuzzy DEA approach for parallelsystems is reviewed. In Sect. 4 the proposed approach to estimate the fuzzy processefficiency of the different processes is presented. Section 5 illustrates the proposedapproach and Sect. 6 summarizes and concludes.
2 Notation
Let us consider n DMUs all of which are structurally homogeneous, i.e. all of themconsist of the same number and type of parallel processes. With no loss of generalitylet us assume that every input is consumed by all the processes and let X
pij denote the
amount of input i consumed by process p of DMU j. Similarly, all processes produceevery output and let Y
pkj denote the amount of output k produced by process p of
DMU j.Unlike in conventional (crisp) network DEA, we will assume that the inputs and
outputs consumed and produced by each process are fuzzy: more specifically, we willassume that they are LR-type Fuzzy Numbers (LRFN) of the form
Xpij =
{(xp
ij
)L,(
xpij
)R,(β
pij
)L,(β
pij
)R}
L,R
Ypkj =
{(yp
kj
)L,(
ypkj
)R,(β
pkj
)L,(β
pkj
)R}
L,R
where it has been assumed, with no loss of generality, the same left and right shapefunctions for all data. These shape functions L,R : [0, 1] → [0, 1] are non-increasing,continuous shape functions and such that L(0) = R(0) = 1 and L(1) = R(1) = 0.
123
Computing fuzzy process efficiency
The membership functions corresponding to these fuzzy sets are of the type
μX(x) =
⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩
1 if (x)L ≤ x ≤ (x)R
L(
(x)L−x(β)L
)if (x)L − (β)L ≤ x ≤ (x)L
R(
x−(x)R
(β)R
)if (x)R ≤ x ≤ (x)R + (β)R
0 otherwise
and the α-cuts of Xpij and Y
pkj are the intervals
(X
pij
)α
=[(
Xpij
)L
α,(
Xpij
)U
α
]
(X
pij
)L
α=
(xp
ij
)L − L−1 (α) ·(β
pij
)L
(X
pij
)U
α=
(xp
ij
)R − R−1 (α) ·(β
pij
)R
(Y
pkj
)α
=[(
Ypkj
)L
α,(
Ypkj
)U
α
]
(Y
pkj
)L
α=
(yp
kj
)L − L−1 (α) ·(β
pkj
)L
(Yp
kj
)U
α=
(yp
kj
)R − R−1 (α) ·(β
pkj
)R
where the inverse shape functions are defined as
L−1 (α) = sup {h : L (h) ≥ α}R−1 (α) = sup {h : R (h) ≥ α}
Let
0 index of specific DMU being assessed
For the multiplier DEA models let
ui multiplier for input ivk multiplier for output k
For the input-oriented envelope DEA models let
θ Radial reduction factor of the inputs consumption of DMU 0λ
pj Intensity variable of process p of DMU j
123
S. Lozano
3 Parallel process approach of Kao and Lin (2012)
Kao and Lin (2012) use Zadeh’s extension principle to compute the system efficiencymembership function as
μE0(e) = sup
x,ymini,k,j
{μXij
(xij),μYkj(ykj)|e = E0(x, y)
}(1)
where E0(x, y) represents the system efficiency of DMU 0 when all the inputs andoutputs are crisp values (x, y). This system efficiency can be computed using themultiplier DEA model
E0(x, y) = max∑
k
∑p
vkypk0
s.t.∑i
∑p
uixpi0 = 1
∑k
vkypkj −
∑i
uixpij ≤ 0 ∀j∀p
ui, vk ≥ 0 ∀i∀k (2)
or, equivalently, its associated dual, the so-called envelope DEA formulation
E0(x, y) = min θ
s.t.∑p
∑j
λpj xp
ij ≤ θ∑
p
xpi0 ∀i
∑p
∑j
λpj yp
kj ≥∑
p
ypk0 ∀k
λpj ≥ 0 ∀j∀p θ free (3)
The lower and upper limits of the α-cuts of the system efficiency (E0)α =[(E0)
Lα , (E0)
Uα
]can be computed solving the following two-level programs
(E0)Uα = max(
Xpij
)L
α∀i∀j∀p
≤ xpij ≤
(Xp
ij
)U
α
(Yp
kj
)L
α∀k∀j∀p
≤ ypkj ≤
(Yp
kj
)U
α
E0(x, y) (4)
123
Computing fuzzy process efficiency
(E0)Lα = min(
Xpij
)L
α∀i∀j∀p
≤ xpij ≤
(Xp
ij
)U
α
(Yp
kj
)L
α∀k∀j∀p
≤ ypkj ≤
(Yp
kj
)U
α
E0(x, y) (5)
The idea is to use the multiplier formulation of the inner program E0(x, y) in the caseof the upper limit (E0)
Uα so that the two programs are of maximization type and can
be merged into one. Analogously, in the case of the lower limit (E0)Lα , the envelope
formulation of E0(x, y, z) is used so that the two programs are of minimization typeand can also be merged into one.
In the end we have the following two non-linear programs (of which the first onecan be appropriately linearized as shown in Appendix A)
(E0)Uα = max
∑p
∑k
vk · ypk0
s.t.∑i
∑p
uixpi0 = 1
∑k
vkypkj −
∑i
uixpij ≤ 0 ∀j∀p
(Xp
ij
)L
α≤ xp
ij ≤(
Xpij
)U
α∀j∀p∀i
(Yp
kj
)L
α≤ yp
kj ≤(
Ypkj
)U
α∀j∀p∀k
ui, vk ≥ 0 ∀i∀k (6)
(E0)Lα = min θ
s.t.∑p
∑j
λpj xp
ij ≤ θ∑
p
xpi0 ∀i
∑p
∑j
λpj yp
kj ≥∑
p
ypk0 ∀k
(Xp
ij
)L
α≤ xp
ij ≤(
Xpij
)U
α∀j∀p∀i
(Yp
kj
)L
α≤ yp
kj ≤(
Ypkj
)U
α∀j∀p∀k
λpj ≥ 0 ∀j∀p θ free (7)
The lower limit model (7) cannot be linearized but since the number of non-linearconstraints is equal to the sum of the number of inputs and outputs, which in mostDEA applications is relatively small, the model should be within the capabilities ofcommercial global optimization solvers like LINGO (Kao and Lin 2012).
123
S. Lozano
In addition to the system efficiency α-cuts, Kao and Lin (2012) propose a methodto estimate the α-cuts of the efficiency of the different parallel process. Specifically,using the optimal solution to model (6) above, the process efficiency upper limits canbe computed as
(Ep
0
)Uα
=∑
k v∗k · (
ypk0
)∗∑
i u∗i · (
xpi0
)∗ (8)
The corresponding process efficiency lower limits can be computed as
(Ep
0
)Lα
=∑
k v∗∗k · (
ypk0
)∗∗∑
i u∗∗i · (
xpi0
)∗∗ (9)
where (u∗∗, v∗∗) are the optimal multipliers corresponding to solving model (6) withthe crisp values (x∗∗, y∗∗) derived from the optimal solution of model (7):
The drawback of this way of computing the process efficiency is that it does not takeinto account that, in general, the optimal multiplier solutions (u∗, v∗) and (u∗∗, v∗∗)used in the above expressions are not unique and therefore the efficiency of eachprocess cannot be determined univocally but varies between certain bounds.
4 Proposed approach for estimating process efficiency
In this section an alternative approach to compute the process efficiencies taking intoaccount this fact is proposed. The first thing we need to formulate are the crisp network
DEA models that compute the bounds Ep+0 (x, y, E0(x, y)) and Ep−
0 (x, y, E0(x, y)) forthe efficiency of a process p compatible with the corresponding system efficiencyE0(x, y). The two models share the same set of constraints and can be expressed as
Ep+0 = max Ep
0 (10)
Ep−0 = min Ep
0 (11)
s.t.
Ep0 =
∑k vk · yp
k0∑i ui · xp
i0∑k vk · yp
kj∑i ui · xp
ij
≤ 1 ∀j∀p
∑p∑
k vk · ypk0∑
p∑
i ui · xpi0
= E0
ui, vk ≥ 0 ∀i∀k
Thus the crisp efficiency of each process p lies within these bounds, i.e.
123
Computing fuzzy process efficiency
Ep−0 (x, yE0(x, y)) ≤ Ep
0(x, y, E0(x, y)) ≤ Ep+0 (x, y, E0(x, y)) (12)
The proposed process efficiency membership function is
μEp
0(e′) = sup
x,ymini,k,j
{μXij
(xij),μYkj(ykj),μE0
(e)
∣∣∣∣∣e = E0(x, y)
Ep−0 (x, y, e) ≤ e′ ≤ Ep+
0 (x, y, e)
}
(13)
and the lower and upper limits of the corresponding α-cuts can be computed solvingthe following two-level programs
(Ep
0
)U
α= max(
Xpij
)L
α∀i∀j∀p
≤ xpij ≤
(Xp
ij
)U
α
(Yp
kj
)L
α∀k∀j∀p
≤ ypkj ≤
(Yp
kj
)U
α
(E0)Lα
≤ E0(x,y) ≤ (E0)Uα
Ep0(x, y, E0(x, y)) (14)
(Ep
0
)L
α= min(
Xpij
)L
α∀i∀j∀p
≤ xpij ≤
(Xp
ij
)U
α
(Yp
kj
)L
α∀k∀j∀p
≤ ypkj ≤
(Yp
kj
)U
α
(E0)Lα
≤ E0(x,y) ≤ (E0)Uα
Ep0(x, y, E0(x, y)) (15)
For the first two-level program the maximum value occurs when E0(x, y) = (E0)Uα
and Ep0(x, y, E0(x, y)) = Ep+
0 (x, y, E0(x, y)), and therefore Ep0(x, y, E0(x, y)) =
Ep+0 (x, y, (E0)
Uα ), which leads to the following one-level optimization model whose
linearization is shown in Appendix B.
(Ep
0
)U
α= max
∑k
vk · ypk0
s.t.∑i
ui · xpi0 = 1
∑k
vk · ypkj −
[∑i
ui · xpij
]≤ 0 ∀j∀p
∑p
∑k
vk · ypk0 − (E0)
Uα ·
⎡⎣∑
p
∑i
ui · xpi0
⎤⎦ = 0
(Xp
ij
)L
α≤ xp
ij ≤(
Xpij
)U
α∀j∀p∀i
123
S. Lozano
(Yp
kj
)L
α≤ yp
kj ≤(
Ypkj
)U
α∀j∀p∀k
ui, vk ≥ 0 ∀i∀k (16)
Similarly, for the second two-level program, the minimum value occurs when
E0(x, y) = (E0)Lα and Ep
0(x, y, E0(x, y)) = Ep−0 (x, y, E0(x, y)), and therefore
Ep0(x, y, E0(x, y)) = Ep−
0 (x, y, (E0)Lα), which leads to the one-level optimization
model
(Ep
0
)L
α= min
∑k
vk · ypk0
s.t.∑i
ui · xpi0 = 1
∑k
vk · ypkj −
[∑i
ui · xpij
]≤ 0 ∀j∀p
∑p
∑k
vk · ypk0 − (E0)
Lα ·
⎡⎣∑
p
∑i
ui · xpi0
⎤⎦ = 0
(Xp
ij
)L
α≤ xp
ij ≤(
Xpij
)U
α∀j∀p∀i
(Yp
kj
)L
α≤ yp
kj ≤(
Ypkj
)U
α∀j∀p∀k
ui, vk ≥ 0 ∀i∀k (17)
The linearization of this model is analogous to that of model (16). This type of LinearPrograms can be solved efficiently using any off-the-shelf optimisation package.
5 Illustration
In this section we will illustrate the proposed approach on a parallel process systemwith two processes both of which consume two inputs and produce a single outputas shown in Fig. 1. The inputs consumed and output produced by each process of the15 DMUs are given by symmetric triangular fuzzy numbers whose α = 0.0 cuts areshown in Table 1. Note that the specific numerical values for this dataset have beenextracted from Ashrafi and Jaafar (2011).
Table 2 shows, for each DMU, the α-cuts of the system efficiency. The correspond-ing upper and lower limits have been computed using models (6) and (7). Note thatfor none of the DMUs the α-cuts contain unity, i.e. no DMU is system efficient. Thatis not surprising since system efficiency in network DEA implies that all processesmust be efficient, something which does not occur often in practice. Thus, as it canbe seen in the following tables, there are some DMUs whose Process 1 is efficient butfor none of these DMUs their corresponding Process 2 is efficient.
123
Computing fuzzy process efficiency
Fig. 1 System with two parallelprocesses 1
1Y
Process 2
21X
22X
21Y
11X
Process 112X
Table 1 Symmetric triangular fuzzy numbers corresponding to inputs and outputs
DMU j Process 1 Process 2(
X11j
)0.0
(X
12j
)0.0
(Y
11j
)0.0
(X
11j
)0.0
(X
12j
)0.0
(Y
11j
)0.0
1 (8,14) (16,19) (13,22) (30,35) (51,58) (21,29)
2 (4,17) (18,22) (19,24) (35,43) (43,48) (15,22)
3 (7,16) (11,26) (18,25) (33,42) (38,45) (21,30)
4 (5,11) (11,24) (12,17) (38,44) (50,61) (12,19)
5 (4,13) (18,24) (13,25) (49,58) (60,66) (10,23)
6 (8,15) (15,21) (12,19) (41,48) (42,49) (22,30)
7 (7,11) (12,29) (18,25) (36,46) (32,38) (12,18)
8 (4,10) (14,25) (14,25) (47,53) (45,53) (18,24)
9 (12,16) (12,23) (11,20) (32,40) (51,58) (12,18)
10 (5,10) (10,20) (20,28) (32,40) (32,39) (19,30)
11 (5,15) (14,27) (11,20) (40,48) (51,61) (21,25)
12 (6,12) (12,23) (20,26) (41,47) (20,41) (9,13)
13 (6,11) (15,26) (21,24) (42,49) (55,62) (17,23)
14 (10,14) (22,26) (19,25) (40,49) (40,50) (11,19)
15 (8,14) (21,27) (21,26) (37,45) (32,39) (10,17)
Tables 3 and 4 show the α-cuts of the efficiency of each process as estimated usingKao and Lin (2012) approach. Tables 5 and 6 show those computed by the proposedapproach. For this problem, the upper limits computed by both methods coincide forboth processes. The lower limit computed by the proposed approach is, however, lowerthan that of Kao and Lin (2012), although not too much lower. The average differencebetween the proposed approach lower limit and that of Kao and Lin (2012) is, forall α values, less than 0.10 in the case of Process 1 and less than 0.05 in the case ofProcess 2.
Figure 2 shows the width of the α-cuts of the system and process efficienciescomputed by Kao and Lin (2012) and by the proposed approach. Note that since thesystem efficiency is the same in both cases the width is also the same. For the processefficiencies, however, the width of the process efficiency estimated by the proposedapproach is greater than that of Kao and Lin (2012). This means that the uncertainty
123
S. Lozano
Tabl
e2
α-c
uts
ofsy
stem
effic
ienc
y
αD
MU
1D
MU
2D
MU
3D
MU
4D
MU
5D
MU
6D
MU
7D
MU
8
(E0)L α
(E0)U α
(E0)L α
(E0)U α
(E0)L α
(E0)U α
(E0)L α
(E0)U α
(E0)L α
(E0)U α
(E0)L α
(E0)U α
(E0)L α
(E0)U α
(E0)L α
(E0)U α
1.0
0.36
90.
369
0.38
20.
382
0.49
00.
490
0.25
70.
257
0.26
40.
264
0.40
80.
408
0.41
10.
411
0.37
00.
370
0.9
0.34
10.
398
0.35
50.
410
0.44
90.
533
0.23
50.
280
0.24
10.
289
0.37
70.
442
0.37
60.
449
0.33
90.
402
0.8
0.31
60.
429
0.33
00.
440
0.41
20.
580
0.21
60.
305
0.21
90.
317
0.34
80.
477
0.34
40.
490
0.31
10.
437
0.7
0.29
10.
462
0.30
60.
472
0.37
70.
630
0.19
70.
332
0.19
80.
346
0.32
10.
514
0.31
40.
535
0.28
50.
475
0.6
0.26
80.
498
0.28
40.
505
0.34
50.
663
0.18
00.
361
0.17
90.
377
0.29
60.
555
0.28
60.
580
0.26
10.
515
0.5
0.24
70.
535
0.26
30.
541
0.31
60.
697
0.16
40.
392
0.16
20.
410
0.27
20.
597
0.26
10.
608
0.23
80.
550
0.4
0.22
70.
573
0.24
30.
575
0.28
80.
731
0.15
00.
426
0.14
50.
446
0.25
00.
643
0.23
70.
637
0.21
70.
579
0.3
0.20
80.
603
0.22
40.
602
0.26
20.
766
0.13
60.
453
0.13
00.
476
0.22
90.
692
0.21
60.
666
0.19
70.
609
0.2
0.19
00.
632
0.20
60.
630
0.23
90.
801
0.12
30.
480
0.11
60.
505
0.20
90.
730
0.19
60.
695
0.17
90.
639
0.1
0.17
30.
658
0.18
90.
653
0.21
70.
837
0.11
20.
508
0.10
30.
534
0.19
10.
766
0.17
70.
725
0.16
20.
669
0.0
0.15
80.
686
0.17
30.
677
0.19
60.
873
0.10
10.
537
0.09
10.
560
0.17
30.
803
0.16
00.
754
0.14
70.
700
αD
MU
9D
MU
10D
MU
11D
MU
12D
MU
13D
MU
14D
MU
15
(E0)L α
(E0)U α
(E0)L α
(E0)U α
(E0)L α
(E0)U α
(E0)L α
(E0)U α
(E0)L α
(E0)U α
(E0)L α
(E0)U α
(E0)L α
(E0)U α
1.0
0.26
50.
265
0.60
00.
600
0.31
50.
315
0.44
30.
443
0.33
60.
336
0.33
50.
335
0.38
90.
389
0.9
0.24
30.
289
0.57
10.
631
0.29
00.
341
0.40
10.
488
0.31
30.
361
0.30
90.
363
0.36
00.
420
0.8
0.22
20.
314
0.54
20.
663
0.26
60.
370
0.36
40.
539
0.29
10.
388
0.28
50.
392
0.33
20.
453
0.7
0.20
30.
342
0.51
40.
695
0.24
50.
401
0.33
00.
575
0.27
00.
416
0.26
30.
423
0.30
70.
488
0.6
0.18
50.
372
0.48
70.
729
0.22
50.
434
0.29
90.
612
0.25
00.
446
0.24
20.
456
0.28
30.
526
0.5
0.16
80.
404
0.46
10.
764
0.20
60.
469
0.27
00.
649
0.23
20.
473
0.22
20.
492
0.26
00.
566
0.4
0.15
30.
432
0.43
60.
801
0.18
90.
508
0.24
50.
687
0.21
40.
496
0.20
40.
530
0.23
90.
609
0.3
0.13
80.
457
0.40
30.
839
0.17
20.
546
0.22
10.
723
0.19
80.
520
0.18
60.
570
0.22
00.
653
0.2
0.12
50.
482
0.36
70.
878
0.15
70.
575
0.20
00.
759
0.18
30.
544
0.17
00.
614
0.20
10.
682
0.1
0.11
30.
508
0.32
70.
919
0.14
30.
604
0.18
00.
797
0.16
80.
569
0.15
50.
646
0.18
40.
711
0.0
0.10
10.
535
0.29
10.
962
0.13
00.
634
0.16
20.
837
0.15
40.
595
0.14
10.
677
0.16
80.
741
123
Computing fuzzy process efficiency
Tabl
e3
Proc
ess
1ef
ficie
ncy
com
pute
dby
Kao
and
Lin
(201
2)ap
proa
ch
αD
MU
1D
MU
2D
MU
3D
MU
4D
MU
5D
MU
6D
MU
7D
MU
8
(E0)L α
(E0)U α
(E0)L α
(E0)U α
(E0)L α
(E0)U α
(E0)L α
(E0)U α
(E0)L α
(E0)U α
(E0)L α
(E0)U α
(E0)L α
(E0)U α
(E0)L α
(E0)U α
1.0
0.62
50.
625
0.67
20.
672
0.72
60.
726
0.51
80.
518
0.56
50.
565
0.53
80.
538
0.65
50.
655
0.62
50.
625
0.9
0.57
40.
679
0.62
50.
721
0.65
30.
808
0.46
70.
575
0.51
30.
622
0.49
20.
588
0.58
90.
730
0.56
20.
695
0.8
0.52
60.
738
0.58
10.
774
0.58
70.
901
0.42
00.
639
0.46
50.
683
0.44
90.
642
0.52
90.
814
0.50
40.
772
0.7
0.48
20.
800
0.54
00.
830
0.52
81.
000
0.37
90.
710
0.42
10.
749
0.41
00.
700
0.47
50.
909
0.45
20.
857
0.6
0.44
00.
867
0.50
00.
889
0.47
51.
000
0.34
10.
789
0.38
00.
820
0.37
30.
763
0.42
71.
000
0.40
50.
952
0.5
0.40
20.
938
0.46
40.
952
0.42
71.
000
0.30
70.
879
0.34
20.
897
0.33
90.
832
0.38
41.
000
0.36
21.
000
0.4
0.36
61.
000
0.42
91.
000
0.38
31.
000
0.27
60.
980
0.30
70.
981
0.30
80.
905
0.34
41.
000
0.32
31.
000
0.3
0.33
21.
000
0.39
61.
000
0.34
41.
000
0.24
81.
000
0.27
51.
000
0.27
90.
985
0.30
91.
000
0.28
81.
000
0.2
0.30
11.
000
0.36
51.
000
0.30
91.
000
0.22
31.
000
0.24
51.
000
0.25
21.
000
0.27
71.
000
0.25
61.
000
0.1
0.27
11.
000
0.33
61.
000
0.27
61.
000
0.20
01.
000
0.21
81.
000
0.22
71.
000
0.24
81.
000
0.22
61.
000
0.0
0.24
41.
000
0.30
81.
000
0.24
71.
000
0.17
91.
000
0.19
31.
000
0.20
41.
000
0.22
21.
000
0.20
01.
000
αD
MU
9D
MU
10D
MU
11D
MU
12D
MU
13D
MU
14D
MU
15
(E0)L α
(E0)U α
(E0)L α
(E0)U α
(E0)L α
(E0)U α
(E0)L α
(E0)U α
(E0)L α
(E0)U α
(E0)L α
(E0)U α
(E0)L α
(E0)U α
1.0
0.55
40.
554
1.00
01.
000
0.47
30.
473
0.82
10.
821
0.68
60.
686
0.57
30.
573
0.61
20.
612
0.9
0.49
50.
618
1.00
01.
000
0.42
30.
528
0.74
70.
903
0.63
10.
746
0.53
30.
615
0.56
90.
658
0.8
0.44
30.
690
1.00
01.
000
0.37
80.
589
0.68
00.
992
0.58
00.
811
0.49
50.
660
0.52
80.
707
0.7
0.39
60.
769
1.00
01.
000
0.33
80.
657
0.61
81.
000
0.53
30.
881
0.45
90.
708
0.48
90.
760
0.6
0.35
30.
858
1.00
01.
000
0.30
10.
733
0.56
21.
000
0.49
00.
958
0.42
60.
759
0.45
30.
816
0.5
0.31
50.
957
1.00
01.
000
0.26
80.
819
0.51
01.
000
0.45
01.
000
0.39
40.
813
0.41
90.
875
0.4
0.28
01.
000
1.00
01.
000
0.23
80.
914
0.46
31.
000
0.41
31.
000
0.36
40.
870
0.38
80.
938
0.3
0.24
81.
000
0.62
31.
000
0.21
21.
000
0.42
01.
000
0.37
81.
000
0.33
60.
931
0.35
81.
000
0.2
0.22
01.
000
0.56
31.
000
0.18
71.
000
0.38
01.
000
0.34
61.
000
0.31
00.
995
0.32
91.
000
0.1
0.19
41.
000
0.49
91.
000
0.16
51.
000
0.34
41.
000
0.31
61.
000
0.28
51.
000
0.30
31.
000
0.0
0.17
11.
000
0.44
01.
000
0.14
61.
000
0.31
11.
000
0.28
81.
000
0.26
11.
000
0.27
81.
000
123
S. Lozano
Tabl
e4
Proc
ess
2ef
ficie
ncy
com
pute
dby
Kao
and
Lin
(201
2)ap
proa
ch
αD
MU
1D
MU
2D
MU
3D
MU
4D
MU
5D
MU
6D
MU
7D
MU
8
(E0)L α
(E0)U α
(E0)L α
(E0)U α
(E0)L α
(E0)U α
(E0)L α
(E0)U α
(E0)L α
(E0)U α
(E0)L α
(E0)U α
(E0)L α
(E0)U α
(E0)L α
(E0)U α
1.0
0.28
70.
287
0.25
40.
254
0.38
40.
384
0.17
50.
175
0.16
40.
164
0.35
70.
357
0.26
80.
268
0.26
80.
268
0.9
0.26
70.
308
0.23
60.
274
0.35
60.
414
0.16
10.
189
0.14
90.
180
0.33
20.
384
0.24
70.
290
0.24
90.
288
0.8
0.24
70.
331
0.21
80.
294
0.32
90.
446
0.14
80.
205
0.13
50.
197
0.30
80.
413
0.22
80.
313
0.23
10.
309
0.7
0.23
00.
355
0.20
20.
316
0.30
40.
480
0.13
50.
222
0.12
20.
215
0.28
50.
443
0.21
00.
337
0.21
40.
332
0.6
0.21
30.
380
0.18
70.
339
0.28
10.
517
0.12
40.
241
0.11
00.
235
0.26
40.
475
0.19
40.
364
0.19
90.
355
0.5
0.19
70.
407
0.17
20.
364
0.25
80.
555
0.11
30.
260
0.09
90.
255
0.24
40.
509
0.17
80.
392
0.18
40.
381
0.4
0.18
10.
436
0.15
90.
390
0.23
80.
596
0.10
40.
281
0.08
80.
278
0.22
50.
545
0.16
30.
422
0.17
00.
408
0.3
0.16
70.
466
0.14
60.
418
0.21
80.
640
0.09
40.
303
0.07
90.
302
0.20
80.
584
0.14
90.
454
0.15
70.
436
0.2
0.15
40.
498
0.13
40.
447
0.20
00.
687
0.08
60.
327
0.07
00.
327
0.19
10.
625
0.13
60.
488
0.14
40.
467
0.1
0.14
10.
532
0.12
20.
478
0.18
30.
737
0.07
80.
353
0.06
20.
354
0.17
50.
668
0.12
40.
524
0.13
20.
499
0.0
0.12
90.
569
0.11
20.
512
0.16
70.
789
0.07
00.
380
0.05
40.
383
0.16
00.
714
0.11
30.
563
0.12
10.
533
αD
MU
9D
MU
10D
MU
11D
MU
12D
MU
13D
MU
14D
MU
15
(E0)L α
(E0)U α
(E0)L α
(E0)U α
(E0)L α
(E0)U α
(E0)L α
(E0)U α
(E0)L α
(E0)U α
(E0)L α
(E0)U α
(E0)L α
(E0)U α
1.0
0.17
20.
172
0.43
10.
431
0.25
70.
257
0.22
50.
225
0.21
40.
214
0.20
80.
208
0.23
80.
238
0.9
0.15
90.
186
0.39
70.
468
0.24
00.
275
0.20
30.
250
0.19
90.
229
0.19
10.
227
0.21
80.
259
0.8
0.14
70.
200
0.36
50.
507
0.22
40.
293
0.18
40.
277
0.18
50.
246
0.17
40.
248
0.20
00.
281
0.7
0.13
60.
215
0.33
50.
549
0.20
90.
313
0.16
60.
307
0.17
20.
263
0.15
90.
270
0.18
20.
306
0.6
0.12
50.
232
0.30
70.
594
0.19
40.
334
0.14
90.
341
0.15
90.
282
0.14
50.
293
0.16
70.
332
0.5
0.11
50.
249
0.28
10.
642
0.18
10.
357
0.13
40.
378
0.14
80.
301
0.13
20.
318
0.15
20.
359
0.4
0.10
60.
267
0.25
60.
694
0.16
80.
381
0.12
10.
420
0.13
70.
322
0.11
90.
345
0.13
80.
389
0.3
0.09
70.
287
0.29
60.
749
0.15
60.
406
0.10
90.
467
0.12
60.
344
0.10
80.
374
0.12
50.
421
0.2
0.08
90.
307
0.27
00.
807
0.14
40.
432
0.09
80.
521
0.11
60.
367
0.09
70.
405
0.11
30.
455
0.1
0.08
10.
329
0.24
10.
870
0.13
30.
460
0.08
80.
581
0.10
70.
392
0.08
80.
439
0.10
20.
492
0.0
0.07
40.
353
0.21
40.
938
0.12
30.
490
0.07
80.
650
0.09
80.
418
0.07
90.
475
0.09
20.
531
123
Computing fuzzy process efficiency
Tabl
e5
Proc
ess
1ef
ficie
ncy
com
pute
dby
prop
osed
appr
oach
αD
MU
1D
MU
2D
MU
3D
MU
4D
MU
5D
MU
6D
MU
7D
MU
8
(E0)L α
(E0)U α
(E0)L α
(E0)U α
(E0)L α
(E0)U α
(E0)L α
(E0)U α
(E0)L α
(E0)U α
(E0)L α
(E0)U α
(E0)L α
(E0)U α
(E0)L α
(E0)U α
1.0
0.62
50.
625
0.67
20.
672
0.72
60.
726
0.51
80.
518
0.56
50.
565
0.53
80.
538
0.65
50.
655
0.62
50.
625
0.9
0.53
70.
679
0.61
00.
721
0.63
30.
808
0.44
90.
575
0.49
10.
622
0.47
70.
588
0.57
30.
730
0.54
70.
695
0.8
0.46
40.
738
0.55
20.
774
0.55
20.
901
0.38
90.
639
0.42
50.
683
0.42
30.
642
0.50
10.
814
0.47
80.
772
0.7
0.40
70.
800
0.50
00.
830
0.48
11.
000
0.33
70.
710
0.36
70.
749
0.37
40.
700
0.43
80.
909
0.41
70.
857
0.6
0.35
70.
867
0.45
20.
889
0.42
01.
000
0.29
20.
789
0.31
50.
820
0.33
00.
763
0.38
31.
000
0.36
30.
952
0.5
0.31
20.
938
0.40
80.
952
0.36
61.
000
0.25
30.
879
0.27
00.
897
0.29
00.
832
0.33
51.
000
0.31
61.
000
0.4
0.27
21.
000
0.36
81.
000
0.31
81.
000
0.21
90.
980
0.23
00.
981
0.25
50.
905
0.29
21.
000
0.27
41.
000
0.3
0.23
71.
000
0.33
11.
000
0.27
71.
000
0.18
91.
000
0.19
51.
000
0.22
40.
985
0.25
51.
000
0.23
71.
000
0.2
0.20
51.
000
0.29
71.
000
0.24
11.
000
0.16
31.
000
0.16
51.
000
0.19
61.
000
0.22
21.
000
0.20
51.
000
0.1
0.17
81.
000
0.26
61.
000
0.20
91.
000
0.14
01.
000
0.13
81.
000
0.17
11.
000
0.19
41.
000
0.17
61.
000
0.0
0.15
31.
000
0.23
81.
000
0.18
11.
000
0.12
01.
000
0.11
51.
000
0.14
91.
000
0.16
81.
000
0.15
11.
000
αD
MU
9D
MU
10D
MU
11D
MU
12D
MU
13D
MU
14D
MU
15
(E0)L α
(E0)U α
(E0)L α
(E0)U α
(E0)L α
(E0)U α
(E0)L α
(E0)U α
(E0)L α
(E0)U α
(E0)L α
(E0)U α
(E0)L α
(E0)U α
1.0
0.55
40.
554
1.00
01.
000
0.47
30.
473
0.82
10.
821
0.68
60.
686
0.57
30.
573
0.61
20.
612
0.9
0.45
30.
618
0.90
51.
000
0.41
30.
528
0.70
70.
903
0.61
70.
746
0.51
40.
615
0.55
10.
658
0.8
0.37
60.
690
0.81
81.
000
0.36
10.
589
0.60
90.
992
0.55
40.
811
0.46
00.
660
0.49
60.
707
0.7
0.32
30.
769
0.74
01.
000
0.31
50.
657
0.52
51.
000
0.49
80.
881
0.41
20.
708
0.44
60.
760
0.6
0.28
10.
858
0.66
81.
000
0.27
40.
733
0.45
31.
000
0.44
70.
958
0.36
80.
759
0.40
10.
816
0.5
0.24
50.
957
0.60
41.
000
0.23
80.
819
0.39
01.
000
0.40
11.
000
0.32
80.
813
0.35
90.
875
0.4
0.21
21.
000
0.54
51.
000
0.20
70.
914
0.33
61.
000
0.35
91.
000
0.29
20.
870
0.32
10.
938
0.3
0.18
31.
000
0.48
11.
000
0.17
91.
000
0.28
91.
000
0.32
21.
000
0.25
90.
931
0.28
71.
000
0.2
0.15
81.
000
0.41
81.
000
0.15
51.
000
0.24
91.
000
0.28
81.
000
0.23
00.
995
0.25
61.
000
0.1
0.13
51.
000
0.35
61.
000
0.13
31.
000
0.21
41.
000
0.25
71.
000
0.20
31.
000
0.22
81.
000
0.0
0.11
51.
000
0.30
21.
000
0.11
51.
000
0.18
31.
000
0.22
91.
000
0.17
91.
000
0.20
31.
000
123
S. Lozano
Tabl
e6
Proc
ess
2ef
ficie
ncy
com
pute
dby
prop
osed
appr
oach
αD
MU
1D
MU
2D
MU
3D
MU
4D
MU
5D
MU
6D
MU
7D
MU
8
(E0)L α
(E0)U α
(E0)L α
(E0)U α
(E0)L α
(E0)U α
(E0)L α
(E0)U α
(E0)L α
(E0)U α
(E0)L α
(E0)U α
(E0)L α
(E0)U α
(E0)L α
(E0)U α
1.0
0.28
70.
287
0.25
40.
254
0.38
40.
384
0.17
50.
175
0.16
40.
164
0.35
70.
357
0.26
80.
268
0.26
80.
268
0.9
0.26
00.
308
0.22
10.
274
0.33
00.
414
0.14
80.
189
0.13
40.
180
0.31
40.
384
0.21
60.
290
0.22
20.
288
0.8
0.23
50.
331
0.19
10.
294
0.28
60.
446
0.12
70.
205
0.11
00.
197
0.27
60.
413
0.17
90.
313
0.18
60.
309
0.7
0.21
20.
355
0.16
50.
316
0.24
90.
480
0.11
20.
222
0.09
20.
215
0.24
30.
443
0.15
10.
337
0.16
30.
332
0.6
0.19
00.
380
0.14
60.
339
0.21
90.
517
0.10
00.
241
0.07
90.
235
0.21
50.
475
0.13
10.
364
0.14
40.
355
0.5
0.17
00.
407
0.13
10.
364
0.19
50.
555
0.08
80.
260
0.06
80.
255
0.19
20.
509
0.11
50.
392
0.12
70.
381
0.4
0.15
20.
436
0.11
80.
390
0.17
30.
596
0.07
80.
281
0.05
70.
278
0.17
20.
545
0.10
00.
422
0.11
20.
408
0.3
0.13
60.
466
0.10
50.
418
0.15
30.
640
0.06
90.
303
0.04
80.
302
0.15
40.
584
0.08
80.
454
0.09
90.
436
0.2
0.12
10.
498
0.09
40.
447
0.13
50.
687
0.06
10.
327
0.04
10.
327
0.13
70.
625
0.07
80.
488
0.08
70.
467
0.1
0.10
70.
532
0.08
40.
478
0.11
90.
737
0.05
30.
353
0.03
40.
354
0.12
20.
668
0.06
80.
524
0.07
70.
499
0.0
0.09
50.
569
0.07
50.
512
0.10
40.
789
0.04
60.
380
0.02
80.
383
0.10
90.
714
0.06
00.
563
0.06
70.
533
αD
MU
9D
MU
10D
MU
11D
MU
12D
MU
13D
MU
14D
MU
15
(E0)L α
(E0)U α
(E0)L α
(E0)U α
(E0)L α
(E0)U α
(E0)L α
(E0)U α
(E0)L α
(E0)U α
(E0)L α
(E0)U α
(E0)L α
(E0)U α
1.0
0.17
20.
172
0.43
10.
431
0.25
70.
257
0.22
50.
225
0.21
40.
214
0.20
80.
208
0.23
80.
238
0.9
0.15
20.
186
0.39
70.
468
0.21
90.
275
0.18
30.
250
0.18
40.
229
0.17
60.
227
0.19
60.
259
0.8
0.13
50.
200
0.36
50.
507
0.18
80.
293
0.15
10.
277
0.16
00.
246
0.15
00.
248
0.16
30.
281
0.7
0.11
90.
215
0.33
50.
549
0.16
60.
313
0.13
00.
307
0.14
40.
263
0.12
80.
270
0.13
80.
306
0.6
0.10
40.
232
0.30
70.
594
0.14
90.
334
0.11
20.
341
0.13
00.
282
0.11
00.
293
0.11
70.
332
0.5
0.09
20.
249
0.28
10.
642
0.13
30.
357
0.09
70.
378
0.11
80.
301
0.09
80.
318
0.10
20.
359
0.4
0.08
00.
267
0.25
60.
694
0.11
80.
381
0.08
40.
420
0.10
80.
322
0.08
70.
345
0.09
10.
389
0.3
0.07
00.
287
0.22
70.
749
0.10
50.
406
0.07
30.
467
0.09
80.
344
0.07
70.
374
0.08
00.
421
0.2
0.06
10.
307
0.19
80.
807
0.09
40.
432
0.06
30.
521
0.08
80.
367
0.06
70.
405
0.07
10.
455
0.1
0.05
30.
329
0.16
80.
870
0.08
30.
460
0.05
50.
581
0.08
00.
392
0.05
90.
439
0.06
30.
492
0.0
0.04
60.
353
0.14
20.
938
0.07
30.
490
0.04
70.
650
0.07
20.
418
0.05
20.
475
0.05
50.
531
123
Computing fuzzy process efficiency
Proposed approach
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
alpha
System Process 1 Process 2
Kao and Lin (2012) approach
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
alpha
System Process 1 Process 2
Fig. 2 Width of α-cuts of system and process efficiencies
in the estimation of the process efficiency is larger and the reason is that the proposedapproach takes into account the inherent variability of the process efficiency even forcrisp data. Note also that, for both methods, the width of the efficiency of Process 1 islarger than that of Process 2.
6 Conclusions
This paper studies the use of DEA to assess the efficiency of parallel process systemswith fuzzy data. The main contribution of the paper is a new approach for estimatingthe individual processes efficiencies. The computational burden is not high since thecomputation of the efficiency of each process requires solving just a couple of easy-to-solve Linear Programs for each α-cut. The rationale behind the proposed approachis that, due to the possibility of alternative optimal weights common to all multiplierDEA models, the process efficiencies are not univocally determined, not even forcrisp data. This increases the uncertainty of the estimations in the fuzzy data case.This expectation has been confirmed experimentally in the parallel-process datasetused for illustration. The increase in uncertainty in the process efficiency estimation,although dependent on the specific process, does not seem to be excessive, however.This empirical observation needs to be confirmed through further research. The same
123
S. Lozano
can be said of the coincidence between the upper limits of the process efficiency com-puted by both approaches. Finally, other fuzzy DEA approaches (e.g. Lertworasirikulet al. 2003a,b,c) could also be extended to the parallel processes context.
Acknowledgments This research was carried out with the financial support of the Spanish Ministry ofScience grant DPI2010-16201, and FEDER.
7 Appendix A
In order to linearize model (6) just consider the new variables
xpij = ui · xp
ij ∀j∀p∀i
ypkj = vk · yp
kj ∀j∀p∀k
Expressing the model with these variables results in the following Linear Program(LP)
(E0)Uα = max
∑p
∑k
ypk0
s.t.∑p
∑i
xpi0 = 1
∑k
vk · ypkj −
∑i
ui · xpij ≤ 0 ∀j∀p (18)
ui ·(
Xpij
)L
α≤ xp
ij ≤ ui ·(
Xpij
)U
α∀j∀p∀i
vk ·(
Ypkj
)L
α≤ yp
kj ≤ vk ·(
Ypkj
)U
α∀j∀p∀k
ui, vk ≥ 0 ∀i∀k
8 Appendix B
In order to linearize model (16) consider the same new variables as in Appendix A.The resulting LP is
(Ep
0
)U
α= max
∑k
ypk0
s.t.∑i
xpi0 = 1
∑k
vk · ypkj −
∑i
ui · xpij ≤ 0 ∀j∀p
123
Computing fuzzy process efficiency
∑p
∑k
ypk0 − (E0)
Uα ·
⎡⎣∑
p
∑i
xpi0
⎤⎦ = 0 (19)
ui ·(
Xpij
)L
α≤ xp
ij ≤ ui ·(
Xpij
)U
α∀j∀p∀i
vk ·(
Ypkj
)L
α≤ yp
kj ≤ vk ·(
Ypkj
)U
α∀j∀p∀k
ui, vk ≥ 0 ∀i∀k
References
Ashrafi, A., & Jaafar, A. B. (2011). Efficiency measurement of series and parallel production systems withinterval data by data envelopment analysis. Australian Journal of Basic and Applied Sciences, 5(11),1435–1443.
Färe, R., & Grosskopf, S. (2000). Network DEA. Socio-Economic Planning Sciences, 34, 35–49.Hatami-Marbini, A., Emrouznejad, A., & Tavana, M. (2011). A taxonomy and review of the fuzzy data
envelopment analysis literature: Two decades in the making. European Journal of Operational Research,214, 457–472.
Kao, C., & Hwang, S. N. (2008). Efficiency decomposition in two-stage data envelopment analysis: Anapplication to non-life insurance companies in Taiwan. European Journal of Operational Research, 185,418–429.
Kao, C., & Hwang, S. N. (2010). Efficiency measurement for network systems: IT impact on firm perfor-mance. Decision Support Systems, 48(3), 437–446.
Kao, C., & Liu, S. T. (2011). Efficiencies of two-stage systems with fuzzy data. Fuzzy Sets and Systems,176, 20–35.
Kao, C. (2012). Efficiency decomposition for parallel production systems. Journal of the OperationalResearch Society, 63, 64–71.
Kao, C., & Lin, P. H. (2012). Efficiency of parallel production systems with fuzzy data. Fuzzy Sets andSystems, 198, 83–98.
Lertworasirikul, S., Fang, S. C., Nuttle, H. L., & Joines, J. A. (2003a). Fuzzy BCC model for data envelop-ment analysis. Fuzzy Optimization and Decision Making, 2, 337–358.
Lertworasirikul, S., Fang, S. C., Joines, J. A., & Nuttle, H. L. W. (2003b). Fuzzy data envelopment analysis(DEA): a possibility approach. Fuzzy Sets and Systems, 139, 379–394.
Lertworasirikul, S., Fang, S. C., Joines, J. A., & Nuttle, H. L. W. (2003c). Fuzzy data envelopment analysis:A credibility approach. In Fuzzy sets based heuristics for optimization. Studies in fuzziness and softcomputing, vol. 126, pp. 141–158.
Lozano, S. (2011). Scale and cost efficiency analysis of networks of processes. Expert Systems With Appli-cations, 38(6), 6612–6617.
Lozano, S., Gutiérrez, E., & Moreno, P. (2013). Network DEA approach to airports performance assessmentconsidering undesirable outputs. Applied Mathematical Modelling, 37(4), 1665–1676.
Tone, K., & Tsutsui, M. (2009). Network DEA: A slacks-based measure approach. European Journal ofOperational Research, 197, 243–252.
123