Dynamic DEA A slacks-based measure approach

Omega 38 (2010) 145 -- 156

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Dynamic DEA: A slacks-based measure approach夡 Kaoru Tonea, ∗ , Miki Tsutsuib a b

National Graduate Institute for Policy Studies, 7-22-1 Roppongi, Minato-ku, Tokyo 106-8677, Japan Central Research Institute of Electric Power Industry, 2-11-2 Iwado Kita, Komae-shi, Tokyo 201-8511, Japan

A R T I C L E

I N F O

Article history: Received 31 October 2008 Accepted 23 July 2009 Processed by Associate Editor Zhu Available online 3 August 2009 Keywords: DEA Dynamic DEA DSBM Carry-over

A B S T R A C T

In data envelopment analysis, there are several methods for measuring efficiency changes over time, e.g. the window analysis and the Malmquist index. However, they usually neglect carry-over activities between two consecutive terms and only focus on the separate time period independently aiming local optimization in a single period, even if these models can take into account the time change effect. In the actual business world, a long time planning and investment is a subject of great concern. For these cases, single period optimization model is not suitable for performance evaluation. To cope with long time point of view, the dynamic DEA model incorporates carry-over activities into the model and enables us to measure period specific efficiency based on the long time optimization during the whole period. Dynamic DEA model proposed by Fa¨ re and Grosskopf is the first innovative contribution for such purpose. In this paper we develop their model in the slacks-based measure (SBM) framework, called dynamic SBM (DSBM). The SBM model is non-radial and can deal with inputs/outputs individually, contrary to the radial approaches that assume proportional changes in inputs/outputs. Furthermore, according to the characteristics of carry-overs, we classify them into four categories, i.e. desirable, undesirable, free and fixed. Desirable carry-overs correspond, for example, to profit carried forward and net earned surplus carried to the next term, while undesirable carry-overs include, for example, loss carried forward, bad debt and dead stock. Free and fixed carry-overs indicate, respectively, discretionary and non-discretionary ones. We develop dynamic SBM models that can evaluate the overall efficiency of decision making units for the whole terms as well as the term efficiencies. © 2009 Elsevier Ltd. All rights reserved.

1. Introduction Measurement of intertemporal efficiency change has long been a subject of concern in DEA. The window analysis by Klopp [15] was the first approach for this purpose (see also Charnes et al. [4]). Based on Malmquist [16], Fa¨ re et al. [9] developed the Malmquist index in the DEA framework. This model can decompose the intertemporal efficiency change into catch-up and innovation (Frontier-shift) effects. However, these models do not account for the effect of carry-over activities between two consecutive terms. For each term these models have inputs and outputs but the connecting activities between terms are not accounted explicitly. The dynamic DEA model proposed by Fa¨ re and Grosskopf [8] is the first innovative scheme for dealing formally with these

夡 Research supported by Grant-in-Aid for Scientific Research, Japan Society for the Promotion of Science. ∗ Corresponding author. Fax: +81 3 6439 6010. E-mail address: [email protected] (K. Tone). 0305-0483/$ - see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.omega.2009.07.003

inter-connecting activities. See also Bogetoft et al. [2], Chen [5], Kao [14], Nemoto and Goto [17,18], Park and Park [19], Sueyoshi and Sekitani [21] and Chang et al. [3] for further references. The dynamic DEA models have close connections with the network DEA models. For the latter subject, refer to Fa¨ re and Grosskopf [8], Tone and Tsutsui [24] and Avkiran [1] among others. In this paper, we extend their model within the slacks-based measure framework proposed by Tone [22] and Pastor et al. [20]. Hence, our model is non-radial and can deal with inputs/outputs individually, enabling us to obtain non-uniform input/output factor efficiencies, contrary to the radial approaches that assume proportional changes in inputs/outputs and provide only one common uniform input/output factor efficiency. We can put weights to input/output items according to their importance. Furthermore, we categorize the carry-overs, called links, into four types, i.e. desirable (good), undesirable (bad), discretionary (free) and non-discretionary (fixed), reflecting actual characteristics of carry-over activities. As for orientations of the model, we have three types: input-, output- and non-oriented. Thus, this paper has the following novel features. (1) As a dynamic model, we can compare a long range performance of companies. (2) Adoption of non-radial SBM models enables us to deal with inputs

146

K. Tone, M. Tsutsui / Omega 38 (2010) 145 – 156

Input t

Term t

Input t+1

Term t +1

Carry-Over t

Output t

Carry-Over t +1

Output t+1

Fig. 1. Dynamic structure.

and outputs individually, and hence their non-proportional changes are allowed. (3) Carry-over activities (links) are categorized into four types, i.e. desirable (good), undesirable (bad), discretionary (free) and non-discretionary (fixed), and hence we can cope with demands of researchers and practitioners correctly and properly. (4) We developed three orientations for every model: input-, output- and nonoriented models. Thus, in accordance with the research purpose, we can choose appropriate models for evaluation. If input (output) side efficiency is the main target, we can choose input-(output-) oriented models. If both input and output efficiencies are to be evaluated concurrently, we can apply non-oriented models. This paper can be positioned as an extension of the network SBM in Tone and Tsutsui [24] to the dynamic structure. The rest of this paper unfolds as follows. In Section 2, we describe our dynamic model (DSBM) and discuss links and their characteristics. The production possibility set and models are presented in Section 3. Objective functions and efficiency of DSBM along with projections are presented in Section 4. We extend our models in Section 5 including the treatment of the initial condition. In Section 6 we exhibit an illustrative example, where we also compare our results with the traditional separation model which deals with time series data term by term. Since uniqueness of the optimal solution is an important matter for identifying term efficiencies and for projecting inefficient DMUs to efficiency status, we report an experiment on this subject using a real world dataset in Section 7. We conclude this paper in the last section. 2. Dynamic structure Traditional DEA models deal usually with efficiency of input resources vs. output products of associated decision making units (DMUs) within cross sectional data. We extend them to dynamic situations as exhibited in Fig. 1. We observe n DMUs over T terms. At each term t, each DMU has its respective inputs and outputs along with the carry-over (link) to the next term t+1. We assume that we have a panel data through terms 1 to T. So, we look at the concerned enterprises as a continuum between the term 1 and the term T.1 What distinguishes dynamic DEA from the ordinary DEA is the existence of carry-overs that connect two consecutive terms. We classify carry-over activities, called links, into four categories as follows:

Comparative shortage of links in this category is accounted as inefficiency. (2) Undesirable (bad) link. This belongs to undesirable carry-over, e.g. loss carried forward, bad debt and dead stock. In our model, undesirable links are treated as inputs and its value is restricted to be not greater than the observed one. Comparative excess in links in this category is accounted as inefficiency. (3) Discretionary (free) link. This corresponds to carry-over that DMU can handle freely. Its value can be increased or decreased from the observed one. The deviation from the current value is not directly reflected in the efficiency evaluation, but the continuity condition between two terms explained in the next section exerts an indirect effect on the efficiency score. However, in Section 4.4, we propose methods for incorporating free link slacks into efficiency account. (4) Non-discretionary (fixed) link. This indicates carry-over that is beyond the control of DMU. Its value is fixed at the observed level. Similarly to free link, fixed link affects the efficiency score indirectly through the continuity condition between two terms. 3. Production possibility set and models We deal with n DMUs (j = 1, . . . ,n) over T terms (t = 1, . . . ,T). At each term, DMUs have common m inputs (i = 1, . . . , m), p nondiscretionary (fixed) inputs (i = 1, . . . , p), s outputs (i = 1, . . . , s) and r non-discretionary (fixed) outputs (i = 1, . . . , r). Let xijt (i = 1, . . . , m), fix

fix

xijt (i = 1, . . . , p), yijt (i = 1, . . . , s)and yijt (i = 1, . . . , r)denote the observed (discretionary) input, non-discretionary input, (discretionary) output and non-discretionary output values of DMU j at term t, respectively. We symbolize the four category links as zgood , zbad , zfree and zfix . In order to identify them by term (t), DMU (j) and item (i), we good employ, for example, the notation zijt (i = 1, . . . , ngood; j = 1, . . . , n; t = 1, . . . , T) for denoting good link values where ngood is the number of good links. These are all observed values up to the term T. fix fix good The production possibilities {xit }, {xit }, {yit }, {yit }, {zit }, {zitbad }, free

fix

{zit } and {zit } are defined by xit ⱖ

n

xijt j

t

(i = 1, . . . , m; t = 1, . . . , T)

fix t

(i = 1, . . . , p; t = 1, . . . , T)

j=1

(1) Desirable (good) link. This indicates desirable carry-over, e.g. retained earnings and net earned surplus carried to the next term. In our model, desirable links are treated as outputs and link value is restricted to be not less than the observed one.

fix

xit =

We discuss the initial condition in Section 5.1.

xijt j

j=1

yit ⱕ 1

n

n j=1

t

yijt j

(i = 1, . . . , s; t = 1, . . . , T)


fix

yit =

n

fix t

yijt j

bad ziot =

(i = 1, . . . , r; t = 1, . . . , T)

j=1

good

zit

ⱕ

n

good t

zijt j

free

(i = 1, . . . , ngood; t = 1, . . . , T)

ziot =

free

n

n

fix

t

bad zijt j

(i = 1, . . . , nbad; t = 1, . . . , T)

fix

ziot =

(i = 1, . . . , nfree; t = 1, . . . , T) fix t

zijt j

n

free

(i = 1, . . . , nfree; t = 1, . . . , T)

n

fix t

zijt j

(i = 1, . . . , nfix; t = 1, . . . , T)

tj = 1 (t = 1, . . . , T)

j=1 free tj ⱖ 0, s− ⱖ 0, s+ ⱖ 0, sgood ⱖ 0, sbad it ⱖ 0 and sit : free(∀i, t), it it it

ⱖ 0 (j = 1, . . . , n : t = 1, . . . , T)

n

free t

zijt j + sit

(i = 1, . . . , nfix; t = 1, . . . , T)

j=1

tj

n

j=1

: free

zit =

(i = 1, . . . , nbad; t = 1, . . . , T)

j=1

j=1

zit

t

bad zijt j + sbad it

j=1

j=1

zitbad ⱖ

n

147

, s+ , s , sbad , and sit are slack variables denoting, respecwhere s− it it it it tively, input excess, output shortfall, link shortfall, link excess and link deviation. good

tj = 1 (t = 1, . . . , T),

(1)

(3)

free

j=1 t

where ∈ Rn (t = 1, . . . , T) is the intensity vector for the term t, and nbad, nfree and nfix are respectively the number of bad, free and fixed links. The last constraint corresponds to the variable returns-toscale assumption (see Cooper et al. [6], pp. 89–94). If we delete this constraint, we have the constant returns-to-scale model. We notice fix fix good bad fix , and zijt on the right of the above are that xijt , xijt , yijt , yijt , zijt , zijt fix

fix

good

observed positive data while xit , xit , yit , yit , zit

free

fix

, zitbad , zit , and zit

tj .2

on the left are variables connected by the intensity variable The continuity of link flows (carry-overs) between terms t and t+1 can be guaranteed by the following condition: n

= zijt j t

j=1

n

zijt j

t+1

(∀i; t = 1, . . . , T − 1),

(2)

j=1

where the symbol stands for good, bad, free or fix. This constraint is critical for the dynamic model, since it connects term t and term t+1 activities.3 Using these expressions for production, we can express DMUo (o = 1, . . . , n) as follows: xiot =

n

xijt j + s− it t

4. Objective functions and efficiency We evaluate the overall efficiency of DMUo (o = 1, . . . , n) takgood free t + }, {sbad ing ({k }, {s− t }, {st }) as variables, in the followt }, {st }, {st ing three orientations, i.e. input-, output- and non-orientations. The input-oriented models deal mainly with reduction of input-related factors while producing at least the observed output-related factor levels. In our DSBM models, we maximize relative slacks in inputs and undesirable (bad) links. In the output-oriented models, we attempt to maximize output-related factors while using no more than the observed amount of any input-related factors. In our DSBM models, we maximize relative slacks in outputs and desirable (good) links. The non-oriented models aim to reduce input-related factors and to enlarge output-related factors simultaneously. Thus, the models in this class unify the input-oriented and output-oriented models in a single framework. These models should be used properly depending on respective research and managerial purposes. The differences in orientations affect the objective function of models as follows. 4.1. Input-oriented case ∗

The input-oriented overall efficiency o is defined by

(i = 1, . . . , m; t = 1, . . . , T)

j=1

fix xiot

=

n

∗o fix t xijt j

(i = 1, . . . , p; t = 1, . . . , T)

j=1

yiot =

n

yijt j − s+ it t

(i = 1, . . . , s; t = 1, . . . , T)

⎡ ⎞⎤ ⎛ T m nbad − sbad w− s 1 1 t⎣ it ⎠⎦ i it ⎝ , = min w 1− + T xiot m + nbad zbad t=1 i=1 i=1 iot

(4)

subject to (2) and (3), where wt and w− are weights to term t and i input i which are supplied exogenously according to their importance and satisfy the conditions as

j=1

fix yiot

=

n

T fix t yijt j

(i = 1, . . . , r; t = 1, . . . , T)

t=1

wt = T

and

m

w− = m. i

(5)

i=1

j=1

good ziot

=

n

good t zijt j

good − sit

(i = 1, . . . , ngood; t = 1, . . . , T)

j=1

2 This expression is an extension of the single term production possibility set explained in Cooper et al. [6, pp. 42–43] to multiple term case. As to the treatments of good (desirable) and bad (undesirable) activities, refer to Cooper et al. [6, pp. 367–369]. 3 We incorporate the initial condition (t = 0) on links in Section 5.1, while the terminal condition (t = T) is already included in (3).

= 1(∀i). If all weights are even, then we can set wt = 1(∀t) and w− i This objective function is based on the input-oriented SBM model [22] and deals with not only excesses in input resources but also undesirable (bad) links as main targets of evaluation. Excesses in undesirable links are accounted in the objective function in the same way as input excesses, because they have similar feature to inputs, i.e. smaller amount is favorable. However, undesirable links are not inputs. They have the role of connection of two consecutive terms as demonstrated by the constraint (2). Each term in square brackets of (4) expresses the efficiency of the term t as measured by the relative

148


slacks of inputs and links, and it is equal to unity if all slacks are zero. It is units-invariant and its value is between 0 and 1. Hence, (4) is the weighted average of term efficiencies over the whole terms which we call the input-oriented overall efficiency and it is also between 0 and 1. t∗ Let an optimal solution (4) subject to (2) and (3) be ({ko }, {s−∗ ot }, }, {sbad∗ {s+∗ ot }, {sot ot }, {sot ∗ efficiency ot by good∗

∗ot

free∗

}). We define the input-oriented term

⎞ ⎛ m nbad sbad∗ w− s−∗ 1 iot i iot ⎠, ⎝ =1− + bad xiot m + nbad ziot i=1 i=1

(t = 1, . . . , T).

(6)

This term efficiency expresses the input-oriented efficiency score ∗ for the term t. The overall efficiency during the period (o ) is the ∗ weighted average of the term efficiencies ot as described below:

∗o =

T 1 t ∗ w ot . T

(7)

t=1

Definition 1. Input-oriented term efficient. ∗ If all optimal solutions of (4) satisfy ot = 1, DMUo is called inputoriented term efficient for the term t. This implies that the optimal slacks for the term t in (6) are all = 0(∀i) and sbad∗ = 0(∀i) for all optimal solutions of (4). zero, i.e. s−∗ iot iot

subject to (2) and (3) where w+ is the weight to output i and satisfies i the condition: s

w+ = s. i

This objective function is an extension of the output oriented SBM model [22] and deals with shortfalls in output products and desirable (good) links as main targets of evaluation. Shortfalls in desirable links are accounted in the objective function in the same way as output shortfalls, because they have similar feature to outputs, i.e. larger amount is favorable. However, desirable links are not outputs. They have the role of connection of two consecutive terms as demonstrated by (2). Each term in square brackets of (8) relates to the efficiency of the term t as measured by the relative slacks of outputs and links, and it is equal to unity if all slacks are zero. It is units-invariant and its value is greater than or equal to 1. Hence, the right of (8) is the weighted average of term efficiencies over the whole terms which is greater than or equal to 1. Since we define the overall efficiency by its reciprocal, the output overall efficiency is between 0 and 1. good∗ t∗ +∗ }, {sbad∗ Using an optimal solution ({ko }, {s−∗ ot }, ot }, {sot }, {sot free∗

{sot }) to (8),(2) and (3) we define the output-oriented term efficiency ∗ot by

∗ot =

Definition 2. Input-oriented overall efficient ∗ If o = 1, DMUo is called input-oriented overall efficient. = 0 (∀i, t) and sbad∗ = 0 (∀i, t). From these Thus, this means s−∗ iot iot definitions, we have: Theorem 1. DMUo is input-oriented overall efficient, if and only if it is input-oriented term efficient for all terms. Furthermore, the input-oriented term efficiencies and the inputoriented overall efficiency satisfy the following Pareto optimality property. Theorem 2. If, for two DMUs a and b, their term efficiencies satisfy the ∗ ∗ ∗ ∗ inequality at ⱖ bt (∀t), then it holds that a ⱖ b . Furthermore, if there ∗ ∗ exists a term t¯ such that at¯ > bt¯ , then we have the strict inequality ∗a > ∗b .

Uniqueness issue 1 ∗ Although the overall efficiency o is uniquely determined as the optimum value of the above objective function (4), the term effi∗ ∗ ciency ot may have multiple optima. If ot has multiple optima, this bad∗ in (6) have multiple optimum solutions and/or s means that s−∗ iot iot and hence projection to the efficiency status described in Section 4.5 cannot be uniquely determined. Consequently, projected input, output and link values may be not unique. This subject is important for the non-radial dynamic SBM models. In order to find its range ∗ of variation, we can solve max (min) ot , while keeping o at the optimum. See Section 7 for an experiment on this issue. 4.2. Output-oriented case The output-oriented overall efficiency ∗o is defined by ⎡ ⎞⎤ ⎛ ngood T s sgood w+ s+ 1 1 t⎣ i it it ⎠⎦ , ⎝ = max w 1+ + good ∗o T yiot s + ngood t=1 i=1 i=1 ziot

1

⎞, good∗ s+∗ ngood siot s w+ 1 i iot ⎠ ⎝ 1+ + i=1 i=1 good yiot s + ngood z ⎛

(t = 1, . . . , T).

iot

(10) The output-oriented overall efficiency during the period (∗o ) is the weighted harmonic mean of the term efficiencies ∗ot as demonstrated below: 1

∗o

=

T 1 wt . T ∗ot

(11)

t=1

For the output-oriented case, we have several definitions and theorems similar to the input-oriented case. See Appendix A. 4.3. Non-oriented case As the combination of input- and output-oriented cases, we define the non-oriented efficiency measure by solving program below:

Proof. From the equality (7), this theorem holds.

1

(9)

i=1

(8)

s− nbad sbad m w− 1 T 1 it i it t 1− + t=1 w i=1 i=1 bad T xiot m + nbad ziot ⎡ ⎞⎤ , ⎛ ∗o = min good + + wi sit sit 1 1 T ngood s ⎠⎦ ⎝ wt ⎣1 + + i=1 i=1 good T t=1 yiot s + ngood z iot

(12) subject to (2) and (3). This objective function is an extension of the non-oriented SBM model [22] and deals with excesses in both input resources and undesirable (bad) links, and shortfalls in both output products and desirable (good) links in a single unified scheme. The numerator is the average input efficiency and the denominator is the inverse of the average output efficiency. We define the non-oriented overall efficiency as their ratio which ranges between 0 and 1, and attains 1 when all slacks are zero. This objective function value is also unitsinvariant. good∗ t∗ +∗ }, {sbad∗ Using an optimal solution ({ko }, {s−∗ ot }, ot }, {sot }, {sot free∗

{sot

}) to (12), (2) and (3) we define the non-oriented term efficiency


as follows:

ot =

We can reify this last constraint by introducing 0–1 integer variables as follows:

w− s−∗ i iot

sbad∗ m 1 iot + nbad 1− i=1 i=1 bad xiot m + nbad ziot ⎞ ⎛ good∗ s+∗ s w+ ngood siot 1 i iot ⎠ ⎝ 1+ + i=1 i=1 good yiot s + ngood ziot

149

free−

(t = 1, . . . , T).

sit

ⱕ Mit and sfree+ ⱕ M(1 − it ), it

(16)

where it (i = 1, . . . , nfree : t = 1, . . . , T) is a 0–1 integer variable and M is a big positive number. free Using this manipulation for sit , we replace the objective function (13) (12) to the following: ⎛ ⎡ ⎞⎤ free− − − bad w s s s 1 1 T nfree m nbad it i it it ⎝ ⎠⎦ wt ⎣1 − + i=1 + i=1 i=1 bad free T t=1 xiot m + nbad + nfree ziot ziot ∗ ⎡ ⎞⎤ . ⎛ (17) o = min good free+ w+ s+ sit sit 1 1 T ngood nfree s i it ⎠⎦ ⎝ wt ⎣1 + + i=1 + i=1 i=1 good free T t=1 yiot s + ngood + nfree z z iot

For the non-oriented case, we have several definitions and theorems similar to the input-oriented case. See Appendix B.

iot

We minimize (17) subject to (2), (3) and (16). We can transform this problem to a 0–1 mixed linear program in the same way as the nonoriented SBM case (see Tone [22]). The new term efficiency can be ∗ re-defined as the components of o in the similar way as (13). We notice that in the occurrence of multiple optima in s−∗ , s+∗ , siot iot iot

good∗

4.4. Incorporation of slacks of free links into efficiency score free

As described in (3), slacks sit free

, sbad∗ iot

free∗

is free in sign, i.e. if siot > 0, then free∗

free

the current value ziot is excessive and if siot < 0, then ziot is in shortage. In the objective functions (4), (8) and (12), slacks in free links are not accounted explicitly, in contrast to the desirable (good) and undesirable (bad) link slacks both of which are included in the respective objective functions. However, in the case that slacks in free links should be accounted in the objective function, we can incorporate them in the following ways: one ex-post way (adjusted score) and the other through 0–1 mixed integer fractional program (MIP). 4.4.1. Ex-post way (Adjusted score) We describe this in the non-oriented model as an example. Once we solve (12) subject to (2) and (3), and obtain an optimal free∗ free link slacks siot (i = 1, . . . , nfree : t = 1, . . . , T). Then we define free∗−

siot

free∗

= max{0, siot } free∗

It holds that siot

free∗− siot

free∗+

and siot free∗−

= siot

free∗+

− siot

free∗

= − min{0, siot }. free∗−

and siot

(14) free∗+

× siot

= 0. Thus,

free∗+ and siot indicate respectively excess and shortfall to the free current value ziot , and they are in an either-or situation. Utilizing them we can redefine the overall efficiency of DMUo ex-post as:

free∗− siot ,

where nfree is the number of free links. The new term efficiency can be re-defined as the components of ¯ ∗o in the similar way as (13). , s+∗ , We notice that in the occurrence of multiple optima in s−∗ iot iot

good∗ siot , sbad∗ iot

free∗ and siot , the new term efficiency ¯ ∗ot

4.5. Projection Let an optimal solution of (4), (8) (12) or (17) subject to (2) and good∗ free∗ 4 t∗ +∗ }, {sbad∗ We define the (3) be ({ko }, {s−∗ ot }, {sot }, {sot ot }, {sot }). projection of DMUo as follows: x¯ iot = xiot − s−∗ iot fix fix x¯ iot = xiot

(i = 1, . . . , p; t = 1, . . . , T)

y¯ iot = yiot + s+∗ iot fix fix y¯ iot = yiot

(i = 1, . . . , m; t = 1, . . . , T)

(i = 1, . . . , s; t = 1, . . . , T)

(i = 1, . . . , r; t = 1, . . . , T)

good good good∗ z¯ iot = ziot + siot bad bad = ziot − sbad∗ z¯ iot iot

(i = 1, . . . , ngood; t = 1, . . . , T)

(i = 1, . . . , nbad; t = 1, . . . , T)

(15)

iot

free free free∗ z¯ iot = ziot − siot fix fix z¯ iot = ziot

(i = 1, . . . , nfree; t = 1, . . . , T)

(i = 1, . . . , nfix; t = 1, . . . , T).

Theorem 3. The projected DMUo is overall-efficient.

4.4.2. The 0–1 mixed integer fractional program (MIP) approach free Since slacks sit of free link variables belong to either excesses or shortfalls, we can express it as a difference of two non-negative variables:

Proof. We prove the theorem in the input-oriented case.

free

free−

= sit

free+

− sit

,

free−

sit

(18)

may not be uniquely

determined.

sit

,

∗ ot

and the new term efficiency may not be uniquely determined. We also notice that the adjusted score is a feasible solution for the MIP solution. Hence, the overall efficiency score of the former is not less than that of the latter.

⎡ ⎞⎤ ⎛ free∗− s−∗ nbad sbad∗ nfree siot m w− 1 1 T iot i iot t ⎠⎦ ⎝ w ⎣1 − + i=1 + i=1 i=1 bad free T t=1 xiot m + nbad + nfree ziot ziot ∗ ⎡ ⎞⎤ , ⎛ ¯ o = good∗ free∗+ s+∗ s w+ ngood siot nfree siot 1 1 T i iot t ⎠⎦ ⎝ w ⎣1 + + i=1 + i=1 i=1 good free T t=1 yiot s + ngood + nfree z z iot

∗it ,

free∗+ siot

free+ ⱖ 0, sfree+ ⱖ 0, sfree− × sit = 0. it it

4 Although we have pointed to the occurrence of multiple optima in the term efficiencies in Section 4.1, we notice here all solutions except the objective value may have multiple optima. In order to find the range of variations of each variable, we must solve max and min of the variable while keeping the objective function value at the optimum. See Section 7 for an experiment.

150


We evaluate the overall-efficiency of the projected DMU. Let an t∗ good∗ ¯ free∗ }, {s¯ bad∗ ot }, {sot }). Then

Input 1

¯ +∗ ¯ optimal solution be ({ko }, {s¯ −∗ ot }, {sot }, {sot we have: x¯ iot =

n

t∗

xijt j + s¯ −∗ iot

(i = 1, . . . , m)

Initial Condition 0

and

j=1 bad z¯ iot

=

n

bad t∗ zijt j

+ s¯ bad∗ iot

Term 1

Carry-Over 1

(i = 1, . . . , nbad).

j=1 bad (i=1, . . . , nbad) by (18), we have: Replacing x¯ iot (i=1, . . . , m) and z¯ ito

xiot =

n

t∗

xijt j + s¯ −∗ + s−∗ iot iot

Output 1

(i = 1, . . . , m) and

Fig. 2. The initial condition.

j=1 bad = zito

n

t∗

bad bad∗ zijt j + s¯ bad∗ iot + siot

(i = 1, . . . , nbad).

j=1

Corresponding to this expression we have the overall-efficiency, ⎡ ⎛ T m w− (s−∗ + s¯ −∗ ) ∗ 1 1 t⎣ i iot iot ⎝ o = w 1− T xiot m + nbad t=1

+

nbad sbad∗ iot i=1

∗

i=1

+ s¯ bad∗ iot

bad ziot

account the initial condition (term t = 0) if needed as follows. As depicted in Fig. 2, DMU utilizes initial input z0 to term 1, where stands for good, bad, free or fix. good bad }, {zfree } and {zfix }. Let the initial input to term 1 be {zij0 }, {zij0 ij0 ij0 Then we have the initial conditions for DMUo (o = 1, . . . , n) as follows: good

zio0 ⱕ

⎞⎤

n

good 1

zij0 j

(i = 1, . . . , ngood)

j=1

⎠⎦ .

¯ bad∗ If any member of ({s¯ −∗ ot }, {sot }) is positive, then it holds that

o < ∗o .

∗ This contradicts the optimality of o . Thus, we have s¯ −∗ ot = 0 (∀t) bad∗ ¯ and sot = 0 (∀t). Hence, the projected DMU is overall-efficient. Similarly, we can prove the theorem in the output-oriented and non-oriented models.

bad zio0 ⱖ

n

1

bad zij0 j

(i = 1, . . . , nbad)

j=1 free

zio0 =

n

free 1

free

zij0 j + si0 ,

free

si0 : free in sign (i = 1, . . . , nfree)

j=1 fix

zio0 =

n

fix 1

zij0 j

(i = 1, . . . , nfix).

(20)

j=1

4.6. Factor efficiency index SBM is a non-radial model, and thus, we can obtain a factor efficiency index (FEI) for each input, output and link, respectively, using the actual (observed) and projected values as follows: FEI =

Actual Data − 1. Projection

(19)

If FEI is zero, the factor is regarded as efficient, while it is inefficient in the case of positive and negative. If an input factor is inefficient, FEI is positive, implying an input excess. The same holds for bad links. If an output factor is inefficient, FEI is negative, implying an output shortfall. The same holds for good links. For free link, FEI can be positive or negative, i.e. excess or shortfall. We notice that FEI is subject to change by the occurrence of multiple optima of respective program. In contrast, in the radial models, proportional changes of input, output and links are assumed, and thus, all FEIs will be same among inputs, outputs and links, respectively, for each term. This is a big difference between radial and non-radial models. 5. Extensions of models

5.2. Sharing resources between links and inputs/outputs In some instances, carry-over from term t and output from term t may share a certain amount of product among them. For example, an enterprise can divide the profit at term t into dividend to share holder (output at t) and retained earnings (carry-over) to t+1. Fig. 3 depicts such situation. Corresponding to such cases, for example, we can relax the congood straints on yiot and ziot in (3) as follows: good

yiot + ziot

=

n good t good (yijt + zijt )j − (s+ + sit ), it

(21)

j=1 good

where i is the number of resources shared among yijt and zijt . Correspondingly, in the case all outputs and good links share with each other, we change the objective function (8) to ⎛ ⎡ ⎞⎤ good T s + + 1 ⎝ wi (sit + sit ) ⎠⎦ 1 t⎣ 1 . (22) = max w 1+ good ∗o T s yiot + ziot t=1 i=1 We can deal with other cases as well.

We discuss two extensions of the fore-mentioned model. 6. An illustrative example 5.1. Initial condition So far we have dealt with homogenous panel data, i.e. each term has the same input-output-link structure.5 However, we can 5

The terminal condition is already included in (3).

We generated a sample dataset as exhibited in Table 1. In this dataset, we assume one input, one output, and one link for four years (T1–T4) regarding eight DMUs (A to H). In other words, they can be respectively labor inputs, products and capital assets which are carried over to the next period. Input amounts are increasing during


Input t

151

Input t+1

Term t

Term t +1

Carry-Over t

Output t

Carry-Over t +1

Output t+1

Fig. 3. Sharing resources between links and outputs.

Table 1 Sample data. DMU

Input

A B C D E F G H

Output

Link

T1

T2

T3

T4

T1

T2

T3

T4

T1

T2

T3

T4

10 30 20 30 30 10 30 20

11 33 22 33 33 11 33 22

12 36 24 36 36 12 36 24

13 39 26 39 39 13 39 26

50 150 50 100 150 100 100 100

50 150 100 120 135 90 180 40

50 150 150 150 120 40 95 150

50 150 180 180 105 35 200 100

10 20 30 15 20 10 20 10

10 25 30 20 25 10 30 10

10 30 30 25 30 10 40 10

10 35 30 30 35 10 50 10

Table 2 Results of dynamic model (adjusted score). Over all efficiency

A B C D E F G H

0.706 0.746 1 0.846 0.567 0.917 0.749 0.851

Table 4 Results of separate model.

Term efficiency

Over all efficiency

T1

T2

T3

T4

0.544 0.683 1 0.842 0.625 1 0.767 0.85

0.833 0.917 1 0.875 0.642 1 0.782 0.778

0.75 0.75 1 0.833 0.558 0.833 0.672 1

0.694 0.635 1 0.833 0.444 0.833 0.775 0.778

A B C D E F G H

0.563 0.593 0.584 0.606 0.505 0.722 0.519 0.771

Term efficiency T1

T2

T3

T4

0.5 0.625 0.208 0.5 0.625 1 0.417 0.75

0.556 0.611 0.463 0.556 0.55 1 0.667 0.333

0.5 0.5 0.667 0.533 0.4 0.4 0.29 1

0.694 0.635 1 0.833 0.444 0.486 0.704 1

(Input-oriented and free link under CRS).

(Input-oriented SBM under CRS).

Table 3 Results of dynamic model (MIP approach).

the input-oriented DSBM with free link under the constant returnsto-scale (CRS) assumption.6 To clarify favorable features of dynamic model, we compare the results with those of separate model, which measures efficiency score separately for each period, i.e. short-term evaluation. In this case, the link variable (capital assets) is treated as input.7 So, in this separation model, no linkage between consecutive terms is accounted. Table 4 is the results for separate model, measured by the input-oriented SBM model under the CRS assumption. Overall efficiency of separate model is calculated as an average of term efficiencies during the four years. In addition, Fig. 4 compares the results of overall efficiency indices, while Fig. 5 depicts rank of these scores. Even though there exist gaps between scores of two dynamic models for DMUs A, B, D and E (Fig. 4), their ranks are completely congruous (Fig. 5). On the other hand, we can find considerable differences between dynamic

Over all efficiency

A B C D E F G H

0.613 0.652 1 0.798 0.539 0.917 0.746 0.851

Term efficiency T1

T2

T3

T4

0.544 0.683 1 0.644 0.625 1 0.767 0.85

0.556 0.611 1 0.783 0.55 1 0.707 0.778

0.611 0.611 1 0.891 0.517 0.833 0.736 1

0.741 0.701 1 0.875 0.464 0.833 0.775 0.778

(Input-oriented and free link under CRS).

the period for all DMUs, while we assumed four settings for output amounts, i.e. stable (A and B), increasing (C and D), decreasing (E and F) and volatile (G and H). Similarly, for link values, we assumed two settings, i.e. stable (A, C, F and H) and increasing (B, D, E and G). The results of dynamic model are shown in Tables 2 and 3, which are the adjusted score explained in Section 4.4.1 and the results of MIP approach in Section 4.4.2, respectively. They are generated by

6 In the MIP approach, we employed M = 10000. For solving MIP, we utilized LINGO 11.0 (branch and bound) and GAMS 2.5 (CPLEX was selected as a solver). They gave the same solution. 7 In a considerable number of previous studies, the capital variable was treated as input (see [13]).

152


1.0

Efficiency Score

0.8 0.6 0.4 0.2 0.0

A

B

C

D

E

F

G

H

Dynamic 1

0.706

0.746

1

0.846

0.567

0.917

0.749

0.851

Dynamic 2

0.613

0.652

1

0.798

0.539

0.917

0.746

0.851

Separate

0.563

0.593

0.584

0.606

0.505

0.722

0.519

0.771

* Dynamic 1: adjusted scores, Dynamic 2: scores by MIP approach Fig. 4. Comparison of overall efficiency indices.

1 2 3 Rank

4 5 6 7 8 9

A

B

C

D

E

F

G

H

Dynamic 1

7

6

1

4

8

2

5

3

Dynamic 2

7

6

1

4

8

2

5

3

Separate

6

4

5

3

8

2

7

1

* Dynamic 1: adjusted scores, Dynamic 2: scores by MIP approach Fig. 5. Comparison of rank of overall efficiency.

and separate models, especially in the rank of overall efficiency. We checked the uniqueness of term efficiencies and found no multiple solutions in this case. To investigate the difference between models in depth, we focus on FEIs of capital assets. Figs. 6–8 are the FEIs of DMUs C, D and G as examples which are measured by Dynamic 1 (adjusted score), Dynamic 2 (MIP) and separate models. We checked the uniqueness of these FEIs and found no multiple solutions. As we mentioned in Section 4.6, the fact that FEI is equal to zero means that the capital investment is efficient, while positive and negative FEIs indicate excess and shortfall, respectively. The outputs of DMU C are assumed to rise steeply with the period, while it stably keeps its relatively large assets from the first term (Table 1). Consequently, it is evaluated as over-invested in the first term in the separate model as indicated in Fig. 6. In contrast to the separate model, the dynamic model evaluates that the large assets in the first term is not only for the production in this term but also for future productions. In other words, DMU C invests on capital assets in advance preparing for the future increases in production demand.

6 Dynamic 1-C Dynamic 2-C Separate-C

5 4 3 2 1 0 T1

T2

T3

T4

Fig. 6. Comparison of FEIs of DMU C.

As a result, both dynamic models certified FEI of DMU C as efficient during the whole period. This example implies that the dynamic model can evaluate DMU activities from the long term view point by


this section we examine uniqueness of optimal solutions using a real world data set.

2 Dynamic 1-D Dynamic 2-D Separate-D

1.5 1

7.1. Model and method

0.5 0 -0.5 -1 T1

T2

T3

T4

Fig. 7. Comparison of FEIs of DMU D.

6 Dynamic 1-G Dynamic 2-G Separate-G

5 4

153

3

We applied both non-oriented and input-oriented DSBM models under the constant and variable returns-to-scale (CRS and VRS) assumptions. We denote the models by CRS, VRS, I-CRS (inputoriented CRS) and I-VRS. We assumed four types of link: free (discretionary), bad (undesirable), good (desirable) and fixed (nondiscretionary) for each model. All these models are solved as a linear programming problem.8 In all cases, the decision variables t good free + }, {sbad are ({ }, {s− t }, {st }, {st t }, {st }). If their optimal values are uniquely determined, all term efficiencies and projection are unique. However, if there exist multiple optima, they may have multiple values. We checked uniqueness of term efficiencies and optimal link values in the following manner.9 In this experiment, we set all = 1(∀i) and w+ = 1(∀i). weights equal to unity, i.e. wt = 1(∀t), w− i i (a) Term efficiency Term efficiencies are individually defined by (6), (10) and (13) for each model. We examined their plurality by solving the Max and Min of ot , ot and ot in terms of the variable + ({ }, {s− }, {sbad t }, {st }). So, for example, in the inputt }, {st }, {st oriented case, we maximize (minimize) t

2

good

free

⎞ ⎛ m nbad sbad w− s− 1 iot i iot ⎠, ⎝ ot = 1 − + xiot m + nbad zbad i=1 i=1 iot

1 0 T1

T2

T3

(t = 1, . . . , T),

(23)

T4 subject to (2) and (3), while keeping

Fig. 8. Comparison of FEIs of DMU G.

accounting for the linkage between consecutive terms represented by links. In the case of DMU D, it keeps relatively small capital assets in the first term, and increases them at slightly greater rate than the rate of output growth. Thus, in the dynamic model, although it is evaluated under-investment in the first term by Dynamic 1, its FEI improved during the period, resulting in being efficient in the final term. The results of Dynamic 2 indicate over-investment unlike Dynamic 1. However, the discrepancy is not so large and the stable smoothed trend is observed. On the other hand, in the separate model, it is evaluated as over-investment in the first three terms and the trend is volatile, especially in the third term. These rapid variations are not uncommon results of the separation model that measures each term efficiency score as completely independent. The case of DMU G is a typical example to demonstrate the volatility of FEI measured by the separate model. Outputs of this DMU fluctuate drastically, while asset investments increase constantly. Since the separate model aims optimization in each term independently, FEI scores of DMU G are volatile sharply reflecting the unstable trend of output production (i.e. production demand). In the actual business, production demand may be highly volatile year by year. However, usually DMUs cannot control capital assets instantly and so it is unreasonable to evaluate its performance annually and independently. In contrast, the dynamic models provide stable FEI scores during the period reflecting the continuity of links (assets) between terms. This implies that dynamic models will produce more practical measures in the output volatility case, too. 7. An experiment on the uniqueness issue of optimal solutions As we mentioned in Section 4.1, non-radial dynamic SBM model may result in multiple optimum solutions for term efficiencies. In

⎛ ⎡ ⎞⎤ T m nbad − sbad w− s 1 t⎣ 1 ∗ it ⎠⎦ i it ⎝ = o . w 1− + bad T xiot m + nbad z t=1 i=1 i=1 iot

(24)

(b) Optimal link value good∗ Optimal link values are determined by means of siot , sbad∗ and iot free∗

siot

in (18). So, for example, in the good link case, we maximize good

(minimize) siot

subject to (2) and (3), while keeping

⎡ ⎞⎤ ⎛ T m sbad w− s− nbad 1 1 t⎣ ∗ it ⎠⎦ i it ⎝ = o . w 1− + T xiot m + nbad zbad t=1 i=1 i=1 iot

(25)

7.2. Data We utilized the data set comprised of 41 US and 9 Japan electric utilities for 7 years: 1990–1996. For each DMU and for each year we used two inputs: the number of employees and consumed fuel in generation division, and a single output: generated power. As the link, we employed the generation capacity as capital assets at the end of each year, which will carry-over to the next year. The Japanese dataset for this study was obtained from [10], while the US dataset was constructed from [7,11,12]. Table 5 exhibits statistics of the dataset. 7.3. Results and observations Since we have 50 DMUs over seven years, we have 350 term efficiencies in total. We checked their plurality by examining the gap of 8 The non-oriented model can be transformed into an equivalent linear program. See Tone [22]. 9 Although we did not check the uniqueness of input and output projections, we can examine them in the same manner.

154


Table 5 Statistics of data set. US

Average S.D. MAX MIN a

JP

Input1 Employee (#)

Input2 Fuel (109 BTUa )

Output Generated power (GWha )

Link Capital assets (MWa )

Input1 Employee (#)

Input2 Fuel (109 BTUa )

Output Generated power (GWha )

Link Capital assets (MWa )

3,044 2,433 12,155 223

328,290 213,425 1,055,926 43,377

32,282 19,884 94,997 5,044

8,557 5,306 24,433 1,487

3,082 2,038 7,375 1,062

656,234 622,821 2,309,568 73,373

72,036 63,178 232,516 14,292

18,031 14,860 53,975 3,956

BTU: British thermal units, GWh: giga watt hour, MW: mega watt.

Table 6 Results of uniqueness check for term efficiencies and optimal link values. Type of link

Model

Term efficiency

Link value

# of non-unique optimal term efficiency (sample = 350)

(%)

Average of (max–min)/max in non-unique (%)

# of non-unique optimal link value (sample = 350)

(%)

Average of (max–min)/max in non-unique (%)

Free

CRS VRS I-CRS I-VRS Average

6 3 0 21 7.5

1.71 0.86 0 6 2.14

0.14 0.1 0 0.48 0.18

49 204 0 238 122.75

14 58.29 0 68 35.07

7.07 17.42 0 13.44 9.48

Bad (undesirable)


3 0 14 14 7.75

0.86 0 4 4 2.21

0.29 0 0.01 0.1 0.1

7 227 11 215 115

2 64.86 3.14 61.43 32.86

0.2 11.77 0.05 9.95 5.49

Good (desirable)


7 0 2 7 4

2 0 0.57 2 1.14

0.19 0 0.49 0.12 0.2

8 199 0 233 110

2.29 56.86 0 66.57 31.43

0.69 21.82 0 11.61 8.53

Fix


7 2 0 0 2.25

2 0.57 0 0 0.64

0.01 0.12 0 0 0.03

0 0 0 0 0

between Max and Min values, i.e. Max–Min. In the case of Max–Min > 0, multiple term efficiencies occur. In Table 6, we report the results corresponding to each link type, i.e. free, bad, good and fixed, and each model, i.e. CRS, VRS, I-CRS and I-VRS.10 On average, nonuniqueness in term efficiency happens in about 2% among 350 cases. In contrast, much more multiple optima exist in the optimal link values. Especially, in the variable returns-to-scale cases (VRS and I-VRS), it amounts to more than 60% on average. It might be caused t by the convexity condition nj=1 j =1, (t =1, . . . , T). We also observed averages of the relative gap, i.e. (Max–Min)/Max, in term efficiencies and links. The results indicate that relatively small variations ( < 0.5%) occur in term efficiencies, whereas large deviations appear in the optimal link values. The small variations in term efficiencies may reflect the fact that all term efficiencies are included in the objective function whose value is uniquely determined as the linear program optimum. Even in the non-oriented models where term efficiencies are not explicitly included in the overall objective function, we observed the same tendency. If the relative gap (Max–Min)/Max

10 In the free link models, we did not take slacks into account in efficiency evaluation in (4), (8) and (12).

0 0 0 0 0

0 0 0 0 0

is comparatively small, we can practically neglect the multiple optima. In this dataset, term efficiencies can be regarded as almost unique from a practical standpoint. However, we notice that this experiment is just for explanatory purpose and does not indicate any general conclusions. Uniqueness of solutions is a very case-dependent matter.

8. Concluding remarks In this paper, we have proposed a slacks-based dynamic DEA (DSBM) model. As an SBM model, DSBM is non-radial and can deal with inefficiency of inputs, outputs and links individually as contrast to the radial models which assume proportional changes of inputs or outputs. Furthermore, we categorized carry-over activities into four types, i.e. good, bad, free and fixed. As the model orientation, we proposed three types, i.e. input-, output- and non-oriented, and offered a program for evaluating the overall and term efficiencies for each orientation. We demonstrated the superiority of the dynamic model over the traditional separation model using an illustrative example. In addition, we also discussed the uniqueness issue of optimal solutions using an actual dataset.


Future research subjects include: (1) decomposition of inefficiency into input, output and link inefficiencies; (2) dynamic cost, revenue and profit efficiencies; (3) dual structure and shadow prices of links; (4) evaluation of Frontier-shift effects in dynamic SBM; (5) further comparisons with other methods, e.g. Window analysis and Malmquist index; and (6) extensions to “Dynamic and Network” SBM models. Recently, Tone [23] proposed new variants of the SBM model which evaluate efficiency of DMUs from the nearest point on the efficient frontier. Application of these new SBM models to dynamic situation would be a future research subject. Acknowledgments We are grateful to two anonymous referees for their valuable comments and suggestions by which our first draft has been substantially improved. Appendix A. On the output-oriented case

Definition 3. (Output-oriented term efficient) If all optimal solutions of (8) satisfy ∗ot = 1, DMUo is called output-oriented term efficient for the term t. This implies that the optimal slacks for the term t in (10) are all good∗ = 0(∀i) and siot = 0(∀i) for all optimal solutions of (8). zero, i.e. s+∗ iot Definition 4. (Output-oriented overall efficient) If ∗o = 1, DMUo is called output-oriented overall efficient. good∗ This means s+∗ = 0(∀i, t) and siot = 0(∀i, t). iot Theorem 4. DMUo is output-oriented overall efficient, if and only if it is output-oriented term efficient for all terms. Furthermore, the output-oriented term efficiencies and the output-oriented overall efficiency satisfy the following Pareto optimality property analogous to Theorem 2. Theorem 5. If, for two DMUs a and b, their term efficiencies satisfy the inequality ∗at ⱖ ∗bt (∀t), then it holds that ∗a ⱖ ∗b . Furthermore, if there exists a term t¯ such that ∗at¯ > ∗bt¯ , then we have the strict inequality ∗a > ∗b . Proof. From (11), we have 1

∗a

=

T T 1 wt 1 wt 1 ⱕ = . T ∗at T ∗bt ∗b t=1

155

This implies that the optimal slacks for term t in (13) are all zero, good∗ = 0(∀i), sbad∗ = 0(∀i), s+∗ = 0(∀i) and siot = 0(∀i) for all optimal i.e. s−∗ iot iot iot solutions of (12). Definition 6. (Non-oriented overall efficient) If the optimal solution for (12) satisfies ∗o = 1, DMUo is called non-oriented overall efficient or briefly overall efficient. good∗ This means s−∗ = 0(∀i, t), sbad∗ = 0(∀i, t), s+∗ = 0(∀i, t) and siot = iot iot iot ∗ 0(∀i, t), and hence ot = 1(∀t). Theorem 6. DMUo is overall efficient, if and only if it is term efficient for all terms. Furthermore, the non-oriented term efficiencies and the nonoriented overall efficiency satisfy the following Pareto optimality property under restricted conditions. Theorem 7. If, for two DMUs a and b, their term efficiencies satisfy the inequality ∗at ⱖ ∗bt (∀t) and furthermore satisfy the conditions: ⎞ ⎛ m nbad sbad∗ w− s−∗ 1 iat i iat ⎠ ⎝ 1− + bad xiat m + nbad ziat i=1 i=1 ⎞ ⎛ m nbad sbad∗ w− s−∗ 1 i iat ibt ⎠, ⎝ ⱖ1 − + bad xibt m + nbad zibt i=1 i=1 ⎞ ⎛ ngood s sgood∗ w+ s+∗ 1 i iat iat ⎠ ⎝ 1+ + good yiat s + ngood i=1 i=1 ziat ⎞ ⎛ s sgood∗ w+ s+∗ ngood 1 i ibt ibt ⎠, ⎝ ⱕ1 + + good yibt s + ngood i=1 i=1 zibt

(t = 1, . . . , T)

(t = 1, . . . , T)

(26)

(27)

then it holds that ∗a ⱖ ∗b . Furthermore, if there exists a term t¯ such that ∗at¯ > ∗bt¯ , then we have the strict inequality ∗a > ∗b . Proof. The conditions (26) and (27) imply ∗at ⱖ ∗bt (∀t) but not vice versa. So (26) and (27) are stronger than the condition ∗at ⱖ ∗bt (∀t). The inequality ∗a ⱖ ∗b can be obtained by comparing components of (12) for a and b. Notice that inequalities (26) and (27) are sufficient conditions for

∗a ⱖ ∗b .11 Uniqueness issue 3 Although the overall efficiency ∗o is uniquely determined as the optimum value of the above fractional program (12), the term efficiency ∗ot may have multiple optima. In order to find its range of variation, we can solve max (min) ∗ot , while keeping ∗o at the optimum. See Section 7 for an example.

t=1

Uniqueness issue 2 Although the overall efficiency ∗o is uniquely determined as the optimum value of the above objective function (8), the term efficiency ∗ot may have multiple optima. In order to find its range of variation, we can solve max (min) ot , while keeping ∗o at the optimum. Appendix B. On the non-oriented case

Definition 5. (Non-oriented term efficient) If all optimal solutions of (12) satisfy ∗ot = 1, DMUo is called non-oriented term efficient or briefly term efficient for the term t.

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11

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