Multiobjective target setting in data envelopment analysis using AHP (original) (raw)

Multiobjective target setting in data envelopment analysis using AHP

Sebastián Lozano*, Gabriel Villa
University of Seville, Sevilla, Spain

Available online 12 October 2007

Abstract

In this paper two new target setting DEA approaches are proposed. The first one is an interactive multiobjective method that at each step of the process asks the decision maker (DM) which inputs and outputs he wishes to improve, which ones are allowed to worsen and which ones should stay at their current level. The local relative priorities of these inputs and outputs changes are computed using the analytic hierarchy process (AHP). After obtaining the candidate target, the DM can update his preferences for improving, worsening or maintaining current inputs and outputs levels and obtain a new candidate target. Thus continuing, until a satisfactory operating point is computed. The second method proposed uses a lexicographic multiobjective approach in which the DM specifies a priori a set of priority levels and, using AHP, the relative importance given to the improvements of the inputs and outputs at each priority level. This second approach requires solving a series of models in order, one model for each priority level. The models do not allow for worsening of neither inputs nor outputs. After the lowest priority model has been solved the corresponding target operating point is obtained. The application of the proposed approach to a port logistics problem is presented. © 2007 Elsevier Ltd. All rights reserved.

Keywords: Data envelopment analysis; Target setting; Analytic hierarchy process; Interactive multiobjective optimization; Lexicographic multiobjective optimization

1. Introduction

Data envelopment analysis (DEA) (e.g. Thanassoulis [1]) has been widely used for evaluating the relative performance of similar decision making units (DMUs). Although, generally, all inputs are considered equally important and all outputs are assumed equally important there are a number of DEA approaches that, in different ways, incorporate the specific preference structure of the decision maker (DM). These approaches generally have one of two different aims: they use the preference information either to derive more effective targets or to provide more meaningful efficiency scores. The former category can be labelled target setting models and the latter efficiency scores models [2].

Among the efficiency scores models, the most common approach of taking into account the DM preference information is through the use of weight constraints. There are many different ways of explicitly restricting weights flexibility [3]. Value efficiency analysis [2,4,5], through the selection of a most preferred solution, is another way of incorporating the DM preference information in order to compute value efficiency scores. This method is also compatible with the use of weight restrictions [6].

[1]

1. • Corresponding author. Fax: +34954487329 .
     E-mail address: slozano@us.es (S. Lozano). ↩︎

We are more interested, however, in target setting models. Of these, the first multiobjective approach was due to Golany [7], which proposed an interactive multi-objective linear programming (MOLP) procedure. Thanassoulis and Dyson [8] proposed a weights-based general preference structure model in which the DM selects a subset of inputs and outputs whose targets should be preferentially improved and specifies weights that reflect the relative importance of such improvements. Also, a pre-emptive weighting structure over the deviation variables is suggested so that higherpriority target levels are met as closely as possible before the achievement of the lower-priority target levels are attempted.

Zhu [9] presented input-oriented, output-oriented and non-oriented DEA models with the objective of maximizing the weighted adjustments in inputs and/or outputs. In these models, inputs and outputs worsening was allowed. Joro et al. [10] compared DEA models with a reference point MOLP approach and presented a lexicographic extension of the reference point model that can exhaust all input and output slacks. Changing the input and output weight vectors (or, better yet, the aspiration level vectors) all efficient operation points can be obtained. Korhonen et al. [1] extended this approach presenting a parameterized general DEA model, which corresponds to a reference direction approach. Korhonen and Syrjänen [12] present an MOLP DEA approach but in a centralized resource allocation context.

Post and Spronk [13] presented an interactive DEA (IDEA) consisting of three steps: select a feasible initial profile, compute potential performance improvement and adjust the selected performance profile. This dynamic process allows for worsening of selected inputs and outputs. They mention that pre-emptive priorities can be incorporated in a sequential (i.e. lexicographic) optimization procedure, although no concrete model was presented. Estellita Lins et al. [14] proposed a multi-objective ratio optimization (MORO) to generate efficient operation points from which the DM may a posteriori choose the one of her preference and, also, an interactive method for multi-objective target optimization (MOTO).

Bogetoft and Nielsen [15] propose interactive benchmarking using a directional distance function approach where the direction vector components can be directly given as relative weights of inputs and outputs or, by subtraction, as aspiration, goal or reference target levels. The direction vector may be restricted, if desired, to input- and outputimproving directions only.

In this paper, two multiobjective DEA approaches for target setting are presented. One of them is an interactive procedure that allows for a progressive articulation of the preferences of the DM. The other is a lexicographic approach that solves a sequence of models that try to improve, in a weighted manner, selected inputs and outputs. Both approaches have in common that they use analytic hierarchy process (AHP) [16] to elicit preference information on the part of the DM.

AHP was recommended as a way of determining the DM preference weights in DEA in Zhu [9]. AHP weights have also been used for determining assurance region weight constraints in DEA models (e.g. Zhu [17], Seiford and Zhu [18], Takamura and Tone [19]). Many papers describe other fruitful collaborations between DEA and AHP by using AHP for aggregating and thus reducing the number of inputs or outputs (e.g. Korhonen et al. [4], Cai and Wu [20]), by using AHP to identify relevant DEA inputs and outputs (e.g. Yoo [21]), by using DEA to derive pairwise comparison values for AHP (e.g. Sinuany-Stern et al. [22]), by using AHP to evaluate qualitative features and derive quantitative inputs or outputs (e.g. Shang and Sueyoshi [23], Yang and Kuo [24], Ertay et al. [25]) or by using DEA to generate relative importance vectors from AHP pairwise comparison matrices and to synthesize global priority weights (e.g. Ramanathan [26]).

The interactive multiobjective approach proposed in this paper is related to the IDEA approach in Post and Spronk [13], the interactive multiobjective procedure in Golany [7], the multiobjective approach in Korhonen et al. [11] and the interactive method in Estellita Lins et al. [14] in that all of them describe a dynamic process. However, the way in which the preference information of the DM is elicited is different. We use AHP to identify and weight the desired changes from the current operating point. Post and Spronk [13] asks the user for aspiration levels after presenting him maximum performance improvements from the current operating point, computed independently along each input and output dimension. Golany [7] presents the DM with a set of alternative targets from which he may choose the one he finds more interesting. Korhonen et al. [11] and Estellita Lins et al. [14] use the Pareto Race interface [27] for letting the DM select his preferred target.

Our interactive multiobjective approach also uses bounds on the inputs and outputs changes and in that sense it is similar to Korhonen and Syrjänen [12] although their approach aims at simultaneously projecting all the DMU within a centralized framework.

The lexicographic approach proposed in this paper is similar to the preemptive weighting structure approach suggested (but not formulated) in Thanassoulis and Dyson [8] and also to the sequential optimization procedure with pre-emptive priorities mentioned (but again not formulated) in Post and Spronk [13]. Apart from mathematically formulating the lexicographic approach, we use a directional distance function that does not exhaust all possible improvements along inputs and outputs with higher priority levels, thus letting some room for improvements along inputs and outputs with lower priority levels. Although not in a lexicographic context, a similar use of the directional distance function for incorporating preferences when searching the frontier can also be found in Bogetoft and Nielsen [15].

The structure of the rest of this paper is the following. In Sections 2 and 3 the proposed interactive and lexicographic multiobjective approaches are, respectively, presented. In Section 4 both approaches are illustrated on a port logistics problem. The last section summarizes and concludes.

2. Interactive multiobjective DEA approach

In this section an interactive multiobjective DEA process is presented. It starts with a given operation point, for example an existing DMU or, better yet, one of its conventional input or output-oriented projections. In any case, the starting operation point will be considered the initial current operation point. An iterative process, described below, is carried out until the DM declares to be satisfied with one of the operation points found. Some additional issues are commented in the second subsection.

2.1. Iterative process

In each iteration tt, assuming a current operation point, the DM is asked to identify the subsets of inputs and outputs for which improvements (and only improvements) are desired ( It−I_{t}^{-}It−and Ot+O_{t}^{+}, respectively) as well as the subsets of inputs and outputs for which no change is allowed ( It0I_{t}^{0} and Ot0O_{t}^{0}, respectively) and, finally, the subsets of inputs and outputs for which eventual worsening may be allowed ( It+I_{t}^{+}It+and Ot−O_{t}^{-}, respectively). All discretionary inputs and outputs must be assigned to any of these six subsets. Next, a simple AHP hierarchy, shown in Fig. 1, is considered. Note that inputs and outputs of a non-discretionary nature, grouped in sets IFI^{\mathrm{F}} and OFO^{\mathrm{F}}, respectively, are not included in the AHP hierarchy nor are those inputs and outputs that belong to sets It0I_{t}^{0} and Ot0O_{t}^{0}, i.e. inputs and outputs whose levels the DM wishes to maintain at their current levels. In the lowest level of the hierarchy the indexes of a couple of inputs and outputs of the corresponding sets ( i,iprimeinIt−i, i^{\prime} \in I_{t}^{-}i,iprimeinIt−and k,kprimeinOt+k, k^{\prime} \in O_{t}^{+}k,kprimeinOt+in the case of improvements and iprimeprime,iprimeprimeprimeinIt+i^{\prime \prime}, i^{\prime \prime \prime} \in I_{t}^{+}iprimeprime,iprimeprimeprimeinIt+and kprimeprime,kprimeprimeprimeinOt−k^{\prime \prime}, k^{\prime \prime \prime} \in O_{t}^{-}kprimeprime,kprimeprimeprimeinOt−in the case of worsening) are shown as an example, i.e. i,i′,i′′,i′′′i, i^{\prime}, i^{\prime \prime}, i^{\prime \prime \prime} are indexes that represent inputs, some of them belonging to It−I_{t}^{-}It−and others to It+I_{t}^{+}It+as specified while, analogously, k,k′,k′′k, k^{\prime}, k^{\prime \prime} and k′′′k^{\prime \prime \prime} are indexes of different outputs some of which belong to Ot+O_{t}^{+}Ot+and others to Ot−O_{t}^{-}Ot−as specified.

In the first level of the hierarchy, the DM is asked to compare the intensity of his improvement desires with that of his worsening tolerance. In this level, the DM establishes the trade-off between the improvements desired in some inputs and outputs and the worsening of other inputs and outputs that he is ready to accept in exchange. While at first glance accepting worse performances in inputs or outputs may not seem a good idea, Pareto efficiency suggests (actually requires) that in order to improve some of the input or output dimensions some others must be allowed to deteriorate.

Fig. 1. AHP hierarchy for interactive multiobjective DEA approach.

Let aimpr, wors a_{\text {impr, wors }} be the pairwise comparison coefficient of the 2×22 \times 2 comparison matrix corresponding to the first-level in the hierarchy of Fig. 1

[1aimpr, wors 1/aimpr, wors 1]\left[\begin{array}{cc} 1 & a_{\text {impr, wors }} \\ 1 / a_{\text {impr, wors }} & 1 \end{array}\right]

The higher the value aimpr, wors a_{\text {impr, wors }}, the higher the preference for the improvements (of the inputs in It−I_{t}^{-}It−and the outputs in Ot+O_{t}^{+}) over the worsening (of the inputs in It+I_{t}^{+}It+and the outputs in Ot−O_{t}^{-}). Thus, a value of aimpr, wors =1a_{\text {impr, wors }}=1 indicates that overall improvement and worsening are given the same importance. The relative importance of both types of changes, improvement and worsening, can be computed as

[wimpr wwors ]=[aimpr, wors 1+aimpr, wors 11+aimpr, wors ]\left[\begin{array}{l} w_{\text {impr }} \\ w_{\text {wors }} \end{array}\right]=\left[\begin{array}{c} \frac{a_{\text {impr, wors }}}{1+a_{\text {impr, wors }}} \\ \frac{1}{1+a_{\text {impr, wors }}} \end{array}\right]

Analogously, for the second level of the hierarchy of Fig. 1:
(a) The inputs and outputs in both It−I_{t}^{-}It−and Ot+O_{t}^{+}Ot+are compared among themselves pairwise with respect to the intensity of the DM’s desire for their eventual improvement. An entry afg>1a_{f g}>1 means that improving input or output ff is more desirable than improving input or output gg and vice versa if afg<1a_{f g}<1.
(b) The inputs and outputs in both It+I_{t}^{+}It+and Ot−O_{t}^{-}Ot−are compared among themselves pairwise with respect to the intensity of the DM’s tolerance for their eventual worsening. An entry apr>1a_{p r}>1 means that worsening of input or output pp is less desirable than worsening of input or output rr and vice versa if apr<1a_{p r}<1.

From these two pairwise comparison matrices, and assuming their consistency is acceptable, valid local relative weights for these inputs and outputs are obtained using the eigenvector method [16]. Let w^it\hat{w}_{i}^{t} and v^kt\hat{v}_{k}^{t} denote the local relative importance of the improving inputs and outputs iinIt−i \in I_{t}^{-}iinIt−and k∈Ot+k \in O_{t}^{+}, respectively and wˉit\bar{w}_{i}^{t} and vˉkt\bar{v}_{k}^{t} denote the local relative importance of the worsening inputs and outputs iinIt+i \in I_{t}^{+}iinIt+and k∈Ot−k \in O_{t}^{-}. The next step is the synthesis, in distributive mode, of the AHP hierarchy to obtain the relative weights witw_{i}^{t} and vktv_{k}^{t} of the inputs and outputs of the four subsets It−,It+,Ot+I_{t}^{-}, I_{t}^{+}, O_{t}^{+}It−,It+,Ot+and Ot−O_{t}^{-}Ot−according to

wit={w^it⋅wimpr if i∈It−,wˉit⋅wwors if i∈It+vkt={v^kt⋅wimpr if k∈Ot+,vˉkt⋅wwors if k∈Ot−w_{i}^{t}= \begin{cases}\hat{w}_{i}^{t} \cdot w_{\text {impr }} & \text { if } i \in I_{t}^{-}, \\ \bar{w}_{i}^{t} \cdot w_{\text {wors }} & \text { if } i \in I_{t}^{+}\end{cases} \quad v_{k}^{t}= \begin{cases}\hat{v}_{k}^{t} \cdot w_{\text {impr }} & \text { if } k \in O_{t}^{+}, \\ \bar{v}_{k}^{t} \cdot w_{\text {wors }} & \text { if } k \in O_{t}^{-}\end{cases}

Since the local relative importance vectors in all levels of the hierarchy are normalized, the sum of the resulting relative weights witw_{i}^{t} and vktv_{k}^{t} is unity.

The final step is the formulation and solution of a DEA model that would seek a feasible operation point (i.e. belonging to the assumed production possibility set) that maximizes the weighted improvements in all inputs and outputs. For the process to move swiftly and in order to achieve effective trade-off among the inputs and outputs improvements and worsening it is advisable to ask the DM to impose some bounds on those changes.

Let xijx_{i j} be the amount of input ii consumed by DMUj,ykj\mathrm{DMU} j, y_{k j} the amount of output kk produced by DMUj,t\mathrm{DMU} j, t the index on current iteration of the interactive process, xit−1x_{i}^{t-1} the amount of input ii consumed in current operation point (initially xi0x_{i}^{0} ), ykt−1y_{k}^{t-1} the amount of output kk produced in current operation point (initially yk0y_{k}^{0} ), Δitt\Delta_{i t}^{t} the maximum improvement amount of input ii from current operation point, Deltait+\Delta_{i t}^{+}Deltait+the maximum worsening amount of input ii from current operation point, δkt+\delta_{k t}^{+} the maximum improvement amount of output kk from current operation point, deltakt−\delta_{k t}^{-}deltakt−the maximum worsening amount of output kk from current operation point, λjt\lambda_{j}^{t} the variables representing the coefficients of a linear combination of existing DMU,xit\mathrm{DMU}, x_{i}^{t} the amount of input ii consumed by new current operation point, ykty_{k}^{t} the amount of output kk produced by new current operation point, αit\alpha_{i}^{t} the proportional reduction of input ii from current operation point and βkt\beta_{k}^{t} the proportional increase of output kk from current operation point.

The DEA model to be solved at each iteration tt, assuming variable returns to scale (VRS), can be formulated as:

Max ∑i∈It−∪It+witαit+∑k∈Ot+∪Ot−vktβkt s.t. ∑jλjt=1∑jλjtxij=xit∀i∉IF,∑jλjtykj=ykt∀k∉OF,xit=(1−αit)⋅xit−1∀i∉IF,ykt=(1+βkt)⋅ykt−1∀k∉OF,xit−1−Δit−⩽xit⩽xit−1+Δit+∀i∉IF,ykt−1−δkt−⩽ykt⩽ykt−1+δkt+∀k∉OF,∑jλjtxij⩽xi0∀i∈IF,∑jλjtykj⩾yk0∀k∈OF,λjt⩾0∀j,xit⩾0,αit free ∀i∉IF,ykt⩾0,βkt free ∀k∉OF.\begin{aligned} & \text { Max } \sum_{i \in I_{t}^{-} \cup I_{t}^{+}} w_{i}^{t} \alpha_{i}^{t}+\sum_{k \in O_{t}^{+} \cup O_{t}^{-}} v_{k}^{t} \beta_{k}^{t} \\ & \text { s.t. } \sum_{j} \lambda_{j}^{t}=1 \\ & \sum_{j} \lambda_{j}^{t} x_{i j}=x_{i}^{t} \quad \forall i \notin I^{\mathrm{F}}, \\ & \sum_{j} \lambda_{j}^{t} y_{k j}=y_{k}^{t} \quad \forall k \notin O^{\mathrm{F}}, \\ & x_{i}^{t}=\left(1-\alpha_{i}^{t}\right) \cdot x_{i}^{t-1} \quad \forall i \notin I^{\mathrm{F}}, \\ & y_{k}^{t}=\left(1+\beta_{k}^{t}\right) \cdot y_{k}^{t-1} \quad \forall k \notin O^{\mathrm{F}}, \\ & x_{i}^{t-1}-\Delta_{i t}^{-} \leqslant x_{i}^{t} \leqslant x_{i}^{t-1}+\Delta_{i t}^{+} \quad \forall i \notin I^{\mathrm{F}}, \\ & y_{k}^{t-1}-\delta_{k t}^{-} \leqslant y_{k}^{t} \leqslant y_{k}^{t-1}+\delta_{k t}^{+} \quad \forall k \notin O^{\mathrm{F}}, \\ & \sum_{j} \lambda_{j}^{t} x_{i j} \leqslant x_{i}^{0} \quad \forall i \in I^{\mathrm{F}}, \\ & \sum_{j} \lambda_{j}^{t} y_{k j} \geqslant y_{k}^{0} \quad \forall k \in O^{\mathrm{F}}, \\ & \lambda_{j}^{t} \geqslant 0 \quad \forall j, \quad x_{i}^{t} \geqslant 0, \quad \alpha_{i}^{t} \text { free } \forall i \notin I^{\mathrm{F}}, \quad y_{k}^{t} \geqslant 0, \quad \beta_{k}^{t} \text { free } \forall k \notin O^{\mathrm{F}} . \end{aligned}

In model (1), it is implicitly assumed that Deltait−=Deltait+=0foralliinIt0Deltait+=0foralliinIt−andandDeltait−=0foralliinIt+.Analogously,.Analogously,deltakt+=deltakt−=0forallkinOt0,deltakt+=0forallkinOt−andanddeltakt−=0forallkinOt+\Delta_{i t}^{-}=\Delta_{i t}^{+}=0 \forall i \in I_{t}^{0} \Delta_{i t}^{+}=0 \forall i \in I_{t}^{-}andand \Delta_{i t}^{-}=0 \forall i \in I_{t}^{+}.Analogously,. Analogously, \delta_{k t}^{+}=\delta_{k t}^{-}=0 \forall k \in O_{t}^{0}, \delta_{k t}^{+}=0 \forall k \in O_{t}^{-}andand \delta_{k t}^{-}=0 \forall k \in O_{t}^{+}Deltait−=Deltait+=0foralliinIt0Deltait+=0foralliinIt−andandDeltait−=0foralliinIt+.Analogously,.Analogously,deltakt+=deltakt−=0forallkinOt0,deltakt+=0forallkinOt−andanddeltakt−=0forallkinOt+. For the non-discretionary inputs and outputs, the constraints only impose that the linear combination projection is not worse than the corresponding initial values.

Note that the LP model (1) above always has a feasible solution since the current operation point is necessarily feasible. That solution has an objective function value equal to 0 since the corresponding change variables αit\alpha_{i}^{t} and βkt\beta_{k}^{t} are 0 for all inputs and all outputs. The current solution may be even optimal, in which case the DM may either decide that the current operation point is satisfactory for her or she may continue the process with new subsets It−,It+,Ot+I_{t}^{-}, I_{t}^{+}, O_{t}^{+}, Ot−,It0O_{t}^{-}, I_{t}^{0} and Ot0O_{t}^{0}.

2.2. Additional issues

About the possibility of the occurrence of the rank reversal problem that may occur in general in AHP (see for example, Wang and Elhag [28]), due to the special structure of the decision hierarchy described above the proposed approach is not vulnerable to this problem. The reason is that in the usual AHP hierarchy all the alternatives in the lowest level of the hierarchy are connected with all the nodes (generally the criteria or subcriteria) in the level above. That does not happen in our case since the lowest level nodes (i.e. the alternatives) consist in the different inputs and outputs whose increase/decrease relative importance is being computed. However, each input or output is connected with only one parent in the hierarchy, which may be the improvement or the worsening node. In any case, although it may happen that the DM inserts a new input or output among the existing alternatives or discards an existing one, since the relative importance values of the affected alternatives are computed solely based on the corresponding pairwise alternatives comparison matrix and the corresponding relative importance vector is computed so that the pairwise comparison values are respected, the relative order between any two alternatives will not change when an alternative is inserted or deleted.

With respect to the iterative process, in a certain iteration the DM may choose It+=Ot−=∅I_{t}^{+}=O_{t}^{-}=\emptyset, i.e. only operation points that weakly dominate the current one are considered. In that case, the corresponding AHP hierarchy is simpler; the

Fig. 2. Flowchart of proposed interactive multiobjective approach.
hierarchy only has one level since the DM has opted for improvement only. Actual improvement will only be obtained if the current operating point is not on the efficient frontier and therefore this type of hierarchy may be used when the DM wants to be assured that the current operating point is non-dominated. Although, in principle, the symmetric case in which the AHP hierarchy only consists of worsening inputs and outputs (i.e. It−=Ot+=∅I_{t}^{-}=O_{t}^{+}=\emptyset ) could also be

conceived, there is no need to consider it since in that case the optimal solution of DEA model (1) would be the current operating point.

Finally note that, if some or all of the successive operation points visited are saved (and not only the most recent current operation point), at any iteration the DM can backtrack and resume the exploration from the one he wishes or, in any case, from the initial operation point. Fig. 2 shows the flowchart that describes the proposed interactive multiobjective approach.

3. Lexicographic multiobjective DEA approach

This section presents a lexicographic multiobjective approach to project an existing DMU J onto the efficient frontier, obtaining appropriate target inputs and outputs. First, the differences with the interactive method are highlighted. In the second subsection, the lexicographic method is described in detail. Finally, additional issues are commented.

3.1. Lexicographic versus interactive method

While the interactive multiobjective approach described in the previous section uses a single DEA model that needs to be solved only once to find an operating point for the consideration of the DM, the lexicographic approach, on the contrary, needs to solve a series of DEA models in order to compute a single target operating point. These models must be solved in order (hence the term lexicographic), according to the priority levels a priori established by the DM. The target is obtained once the model corresponding to the lowest priority level has been solved.

Another very important difference between both approaches is that the interactive multiobjective approach is iterative by nature and assumes that the DM articulates his preferences progressively along the process, in a dynamic way, while the lexicographic multiobjective method requires an a priori articulation of preferences. Therefore, this approach should only be used when the DM is able (and willing) to state his preferences a priori.

Since this requirement can be considered as informational handicap it may be argued that this approach has less practical appeal than the interactive multiobjective approach. We consider it however for completeness, because this preemptive-priority type of approach has been suggested in the literature by several authors but never mathematically formulated, and also because from a theoretical standpoint it is a perfectly valid approach.

A final difference with the proposed interactive multiobjective approach is that in the lexicographic multiobjective approach only inputs and outputs improvements are considered, i.e. the target operation point must dominate the DMU being projected. Otherwise, high priority inputs and outputs may improve “too much” at the expense of the low priority ones. Constraining all inputs and outputs to maintain their current levels at least neutralizes that risk.

3.2. Process description

The first decision the DM must make is to establish the number of priority levels LL and the subsets of inputs and outputs considered at each priority level t,Itt, I^{t} and OtO^{t}, respectively. If the DM does not consider an input or output as candidate for improvement (for example, because it is non-discretionary) then that input or output should not appear in any of the ItI^{t} and OtO^{t} subsets. Let IFI^{\mathrm{F}} and OFO^{\mathrm{F}} represent the subsets containing such inputs and outputs, respectively.

Fig. 3. AHP hierarchy for lexicographic multiobjective DEA approach.

For each priority level tt, the AHP hierarchy shown in Fig. 3 is used to obtain relative weights for the improvement of the inputs and outputs considered in that priority level. Note that the first level of the hierarchy asks the DM to establish a trade-off between improving the inputs selected in that priority level versus improving the outputs also assigned to that priority level. In the lowest level of the hierarchy the indexes of four inputs and outputs of the corresponding sets (i,i′,i′′,i′′′∈It\left(i, i^{\prime}, i^{\prime \prime}, i^{\prime \prime \prime} \in I^{t}\right. and k,k′,k′′,k′′′∈Otk, k^{\prime}, k^{\prime \prime}, k^{\prime \prime \prime} \in O^{t} ) are shown as an example, i.e. i,i′,i′′,i′′′i, i^{\prime}, i^{\prime \prime}, i^{\prime \prime \prime} are indexes that represent inputs that belong to ItI^{t} and k,k′,k′′k, k^{\prime}, k^{\prime \prime} and k′′′k^{\prime \prime \prime} are indexes of the different outputs that belong to OtO^{t}.

Note also that for those priority levels composed exclusively of outputs (i.e. It=∅I^{t}=\emptyset ) or inputs (i.e. Ot=∅O^{t}=\emptyset ), the corresponding AHP hierarchy simplifies to a single level. These special cases resemble what traditionally are termed an input and output orientation, respectively, i.e. giving all the preference to improving the inputs (although only the selected ones in our case) or to improving the outputs.

Returning to the general case of Fig. 3, in the first level of the hierarchy, the DM is asked to compare the intensity of his desire for inputs reduction versus outputs increase. Let ainput, output a_{\text {input, output }} be the pairwise comparison coefficient of the 2×22 \times 2 comparison matrix corresponding to the first-level in the hierarchy

[1ainput, output 1/ainput, output 1]\left[\begin{array}{ll} 1 & a_{\text {input, output }} \\ 1 / a_{\text {input, output }} & 1 \end{array}\right]

The higher the value ainput, output a_{\text {input, output }}, the higher the preference for the overall improvement of the inputs in ItI^{t} over the overall improvement of the outputs in OtO^{t}. Thus, a value of ainput, output =1a_{\text {input, output }}=1 indicates that overall improvement of both inputs and outputs are given the same importance. The relative importance of both types of improvements can be computed.

[winput woutput ]=[ainput, output 1+ainput, output 11+ainput, output ]\left[\begin{array}{l} w_{\text {input }} \\ w_{\text {output }} \end{array}\right]=\left[\begin{array}{c} \frac{a_{\text {input, output }}}{1+a_{\text {input, output }}} \\ \frac{1}{1+a_{\text {input, output }}} \end{array}\right]

Analogously, in the second level of the AHP hierarchy the relative importance of improving the different individual inputs (respectively, outputs) within each set ItI^{t} and OtO^{t} is compared, i.e.
(a) The inputs in ItI^{t} are compared among themselves pairwise with respect to the intensity of the DM’s desire for their eventual improvement. An entry aii′>1a_{i i^{\prime}}>1 means that reducing input ii is more desirable than reducing input i′i^{\prime} and vice versa if aii′<1a_{i i^{\prime}}<1.
(b) The outputs in OtO^{t} are compared among themselves pairwise with respect to the intensity of the DM’s desire for their eventual improvement. An entry akk′>1a_{k k^{\prime}}>1 means that increasing output kk is more desirable than increasing output k′k^{\prime} and vice versa if akk′<1a_{k k^{\prime}}<1.

From these two pairwise comparison matrices, and assuming their consistency is acceptable, valid local relative importance for these inputs and outputs are obtained using the eigenvector method [16]. Let w^it\hat{w}_{i}^{t} and v^kt\hat{v}_{k}^{t} denote these local relative importances. The next step is the synthesis, in distributive mode, of the AHP hierarchy to obtain the relative weights witw_{i}^{t} and vktv_{k}^{t} of the inputs and outputs of the two subsets ItI^{t} and OtO^{t}

wit=w^it⋅winput ,vkt=v^kt⋅woutput w_{i}^{t}=\hat{w}_{i}^{t} \cdot w_{\text {input }}, \quad v_{k}^{t}=\hat{v}_{k}^{t} \cdot w_{\text {output }}

As in the interactive case, since the local relative importance vectors in all levels of the hierarchy are normalized, the sum of the resulting relative weights witw_{i}^{t} and vktv_{k}^{t} is unity.

The lexicographic approach involves the solution of a number of DEA models, one for each priority level. Let λjt\lambda_{j}^{t} be the coefficients of linear combination of existing DMU at priority level t,xitt, x_{i}^{t} the amount of input ii consumed by solution obtained at priority level t,yktt, y_{k}^{t} the amount of output kk produced by solution obtained at priority level t,xi0=xiJt, x_{i}^{0}=x_{i J} the amount of input ii initially consumed by DMU being projected, yk0=ykJy_{k}^{0}=y_{k J} the amount of output kk initially produced by DMU being projected, and γt\gamma^{t} the improvement size along the direction given by priority weights of level tt.

The proposed DEA model for priority level tt, assuming VRS, is:

Max γt s.t. ∑jλjt=1∑jλjtxij⩽(1−γtwit)⋅xi0∀i∈It∑jλjtxij⩽[1−(γs)∗wis]⋅xi0∀1⩽s<t∀i∈Is∑jλjtxij⩽xi0∀i∈(⋃s=t+1LIs)∪IF,∑jλjtykj⩾(1+γtvkt)⋅yk0∀k∈Ot,∑jλjtykj⩾[1+(γs)∗vks]⋅yk0∀1⩽s<t∀k∈Os,∑jλjtykj⩾yk0∀k∈(⋃s=t+1LOs)∪OF,λjt⩾0∀jγt⩾0.\begin{aligned} & \text { Max } \gamma^{t} \\ & \text { s.t. } \sum_{j} \lambda_{j}^{t}=1 \\ & \sum_{j} \lambda_{j}^{t} x_{i j} \leqslant\left(1-\gamma^{t} w_{i}^{t}\right) \cdot x_{i}^{0} \quad \forall i \in I^{t} \\ & \sum_{j} \lambda_{j}^{t} x_{i j} \leqslant\left[1-\left(\gamma^{s}\right)^{*} w_{i}^{s}\right] \cdot x_{i}^{0} \quad \forall 1 \leqslant s<t \quad \forall i \in I^{s} \\ & \sum_{j} \lambda_{j}^{t} x_{i j} \leqslant x_{i}^{0} \quad \forall i \in\left(\bigcup_{s=t+1}^{L} I^{s}\right) \cup I^{\mathrm{F}}, \\ & \sum_{j} \lambda_{j}^{t} y_{k j} \geqslant\left(1+\gamma^{t} v_{k}^{t}\right) \cdot y_{k}^{0} \quad \forall k \in O^{t}, \\ & \sum_{j} \lambda_{j}^{t} y_{k j} \geqslant\left[1+\left(\gamma^{s}\right)^{*} v_{k}^{s}\right] \cdot y_{k}^{0} \quad \forall 1 \leqslant s<t \quad \forall k \in O^{s}, \\ & \sum_{j} \lambda_{j}^{t} y_{k j} \geqslant y_{k}^{0} \quad \forall k \in\left(\bigcup_{s=t+1}^{L} O^{s}\right) \cup O^{\mathrm{F}}, \\ & \lambda_{j}^{t} \geqslant 0 \quad \forall j \gamma^{t} \geqslant 0 . \end{aligned}

The objective function represents the maximum improvement (w.r.t. initial inputs and outputs) along the direction given by the weight vectors at this priority level. The constraints on the inputs and output targets computed at this level guarantee that:

• the target levels obtained for inputs and outputs already considered in higher priority levels are maintained;
• for the inputs and outputs not considered in higher priority levels as well as for those belonging to IFI^{\mathrm{F}} and OFO^{\mathrm{F}} the initial input and output levels cannot be worsened.

3.3. Additional issues

Note that the models of all priority levels always have feasible solutions since the initial solution is feasible in the first priority model and the optimum of the previous priority level is always feasible in the next priority level model. Also, for the same reasons stated in the interactive approach, the proposed hierarchy is not vulnerable to the rank reversal problem that may occur in AHP in general. Finally, Fig. 4 shows the flowchart that describes the proposed lexicographic multiobjective approach.

4. Illustration

In this section, the two multiobjective approaches proposed above are illustrated on a real port logistics problem in which the DM was the Port Authority of Seville (PAS). There are a number of studies in the literature that propose the use of DEA for port and container terminals efficiency measurement (e.g. Roll and Hayuth [29], Tongzon [30], Cullinane et al. [31], Barros [32], Ríos and Maçada [33], Cullinane and Wang [34], etc.). After reviewing this literature, four inputs were selected, namely total terminal area (TAREA), total berth length (TBL), number of equivalent cranes (NEC) and number of tugs (NT). Three outputs were considered, namely total cargo traffic (TON), container throughput (TEU) and number of ship calls (NSC). Table 1 shows these data for 27 Spanish ports and for year 2002. More details can be found in Quesada Ibargüen [35].

The starting point considered corresponds to the input and output data of the Port of Seville. As it can be seen in Table 1, these values are TTA =4788909=4788909, TBL =4442=4442, NEC =51.1,NT=2,TON=4704553,TEU=100960.5=51.1, \mathrm{NT}=2, \mathrm{TON}=4704553, \mathrm{TEU}=100960.5

Fig. 4. Flowchart of proposed lexicographic multiobjective approach.
and NSC=1404\mathrm{NSC}=1404. The PAS considered that TTA and TBL (i.e. the land area available and the total quay length) were fixed inputs and therefore it was decided to treat them as non-discretionary inputs, i.e. IF={TTA,TBL}I^{\mathrm{F}}=\{\mathrm{TTA}, \mathrm{TBL}\}. The software package used to solve all the LP models was LINGO version 8.0.

Table 1
Inputs and outputs data of 27 Spanish ports for year 2002

4.1. Interactive multiobjective projection

After explaining to the DM the AHP methodology and the aim of model (1) (avoiding the most technical details) he produced the following inputs/outputs increase/decrease sets: It+={NEC,NT},Ot+={TON,TEU,NSC}I_{t}^{+}=\{\mathrm{NEC}, \mathrm{NT}\}, O_{t}^{+}=\{\mathrm{TON}, \mathrm{TEU}, \mathrm{NSC}\} and It−=Ot−=It0=Ot0=∅I_{t}^{-}=O_{t}^{-}=I_{t}^{0}=O_{t}^{0}=\emptyset. Fig. 5 shows the corresponding pairwise comparison matrices and resulting priority weights.

The DM set the limits on the increases of NEC to 10, of TON to 1 million, of TEU to 200000 and of NSC to 3000. Model (1) was solved and the following operating point was delivered to the DM: TTA =4788909=4788909, TBL =4442=4442, NEC =51.1,NT=3,TON=5704553,TEU=300960.5=51.1, \mathrm{NT}=3, \mathrm{TON}=5704553, \mathrm{TEU}=300960.5 and NSC =4404=4404. All three output increases bounds were binding. It seemed that current operating point was far from the efficient frontier (output technical efficiency had been found to be low, just 62%62 \% ). While NT increases in 1 tug it seems that no more NEC are required. In a second iteration, the DM produced the following inputs/outputs increase/decrease sets: It+={NEC},Ot+={TON,TEU},It0={NT}I_{t}^{+}=\{\mathrm{NEC}\}, O_{t}^{+}=\{\mathrm{TON}, \mathrm{TEU}\}, I_{t}^{0}=\{\mathrm{NT}\}, Ot0={NSC}O_{t}^{0}=\{\mathrm{NSC}\} and It−=Ot−=∅I_{t}^{-}=O_{t}^{-}=\emptyset. Fig. 6 shows the corresponding pairwise comparison matrices and resulting priority weights.

The DM set the limits on the increases of NEC to 10, of TON to 1 million and of TEU to 50000 . Model (1) was solved again and the following operating point was delivered to the DM: TTA =4788909=4788909, TBL =4442=4442, NEC =51.1=51.1, NT=3,TON=6704553,TEU=350960.5\mathrm{NT}=3, \mathrm{TON}=6704553, \mathrm{TEU}=350960.5 and NSC=4404\mathrm{NSC}=4404. All three output increases bounds were again binding and again NEC did not change, which suggested that perhaps they might be reduced somewhat. NT was allowed to increase again. TEU was considered fine but NSC was allowed to decrease if necessary. Thus, in the third iteration, the DM produced the following inputs/outputs increase/decrease sets: It−={NEC},Ot+={TON},Ot−={NSC}I_{t}^{-}=\{\mathrm{NEC}\}, O_{t}^{+}=\{\mathrm{TON}\}, O_{t}^{-}=\{\mathrm{NSC}\}, It+={NT},Ot0={TEU}I_{t}^{+}=\{\mathrm{NT}\}, O_{t}^{0}=\{\mathrm{TEU}\} and It0=∅I_{t}^{0}=\emptyset. Fig. 7 shows the corresponding pairwise comparison matrices and resulting priority weights.

The DM set the limits on the decreases of NEC and NSC to 5 and 200, respectively and the limits on the increases of NT and TON to 1 and 500000 , respectively. Model (1) was solved once more and the following operating point,

Fig. 5. Comparison matrices and relative weights in 1st interactive iteration.

Fig. 6. Comparison matrices and relative weights in 2nd interactive iteration.
which eventually is satisfactory, was delivered to the DM: TTA =4788909=4788909, TBL =4442,NEC=46.1,NT=3=4442, \mathrm{NEC}=46.1, \mathrm{NT}=3, TON =7204553,TEU=350960.5=7204553, \mathrm{TEU}=350960.5 and NSC =4404=4404. Table 2 summarizes the inputs and outputs of the different operation points explored. For comparison, the output-oriented BCC (Banker et al. [36]) projection is also shown.

Fig. 7. Comparison matrices and relative weights in 3rd interactive iteration.

Table 2
Interactive multiobjective targets and BCC-O projection

4.2. Lexicographic multiobjective projection

Since this approach does not allow for input increases nor for output decreases, the targets computed in this case are not comparable with the ones computed with the interactive approach. After explaining this and the working of the lexicographic approach to the DM he decides to establish L=2L=2 priority levels. Since the DM is not interested in reducing NT this input is not considered from improvement. Thus IF={TTA,TBL,NT}I^{\mathrm{F}}=\{\mathrm{TTA}, \mathrm{TBL}, \mathrm{NT}\}. In the first priority level, the remaining input and two of the outputs are considered, namely I1={NEC},O1={TON,TEU}I^{1}=\{\mathrm{NEC}\}, O^{1}=\{\mathrm{TON}, \mathrm{TEU}\}. Finally, the second priority level consists of the third output only, i.e. I2=∅I^{2}=\emptyset and O2={NSC}O^{2}=\{\mathrm{NSC}\}. Fig. 8 shows the pairwise comparison matrices and resulting priority weights for both priority levels.

The solution of the DEA model for the first priority level gives (γ1)∗=1.313\left(\gamma^{1}\right)^{*}=1.313 and an operation point TTA =4788909=4788909, TBL=4442,NEC=9.8,NT=2,TON=7644948,TEU=293669.5\mathrm{TBL}=4442, \mathrm{NEC}=9.8, \mathrm{NT}=2, \mathrm{TON}=7644948, \mathrm{TEU}=293669.5 and NSC=1404\mathrm{NSC}=1404 while the solution of the second priority level gives (γ2)∗=1.415\left(\gamma^{2}\right)^{*}=1.415 and a final target TTA=4788909,TBL=4442,NEC=9.8,NT=2,TON=7644948\mathrm{TTA}=4788909, \mathrm{TBL}=4442, \mathrm{NEC}=9.8, \mathrm{NT}=2, \mathrm{TON}=7644948, TEU=293669.5\mathrm{TEU}=293669.5 and NSC=3390.6\mathrm{NSC}=3390.6, which on this occasion coincides with the BCC-O projection. Although this does not need to happen always, it is clear that, since it can consider improvement but no worsening of inputs and outputs, the lexicographic approach is less flexible than the interactive approach. This limitation could, however, be partially overcome starting with a different, appropriate operation point.

Fig. 8. Comparison matrices and priority weights in lexicographic approach (L=2)(L=2).

4.3. DM feedback

Although the DM was able to apply both methods without much trouble, he found the interactive approach more useful for several reasons. One was the possibility to generate successive operation points, instead of just one. Another was that the AHP hierarchy of the interactive method gave more options, in particular inputs can be increased and outputs can be decreased. Also, the use of bounds on the changes of inputs and outputs was greatly acknowledged by the DM because it allowed him to move in controllable steps. Another handicap of the lexicographic approach was that it required the DM to specify a priori his preferences, which was harder for the DM to do. On the contrary, the interactive method allowed the DM to modify his change preferences (i.e. the change direction) from any iteration to the next, until a satisfactory solution was found. That the DM favored the interactive method is not surprising and coincides with the general trend in MCDM to design interactive methods that can be more effective than other methods.

5. Summary and conclusions

In this paper, two different multiobjective DEA target-setting approaches have been proposed. The first one is an interactive approach in which, at each iteration, the DM projects the current operation point, improving some selected

inputs and outputs, maintaining others and allowing others to worsen, all in a controlled manner (i.e. using weights and bounds). The relative weights of the inputs and outputs to improve or that can worsen at each iteration are computed from a simple AHP hierarchy.

AHP is also used to derive the relative weights of inputs and outputs in the second approach, in which the DM establishes a number of priority levels and assigns the inputs and outputs he wishes to improve to one of these priority levels. A simple AHP hierarchy involving only the corresponding inputs and outputs is solved for each priority level and the corresponding weights are used in the objective function of a DEA model that maximizes, using a directional distance function approach, the improvement of those inputs and outputs. The lexicographic nature of this approach guarantees that the targets computed for the inputs and outputs of each priority level are maintained in the successive, lower priority levels. The target found by proposed lexicographic approach is constrained to dominate the DMU being projected. The two approaches have been illustrated with a real problem on port logistics performance, showing their versatility and their usefulness in finding satisfactory operation points.

Acknowledgements

The authors would like to thank two anonymous referees for their comments and suggestions, which have helped to improve the paper.

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