A Stochastic Bilevel DEA-Based Model for Resource Allocation ^†

Abstract

The optimal allocation of limited resources along with output target setting are critical in pursuing the sustainability and competitiveness of organizations. The process of resource distribution is usually implemented through a central unit that routes resources to the subordinate decision-making units (DMUs) along with DMUs lower bounds of desired efficiency. Moreover, the central unit has the authority to set the overall expected output targets so as to maximize organizational effectiveness. In this paper, we tackle evaluation efficiency questions using a type of bilevel network data envelopment analysis (DEA) approach within a stochastic framework. The proposed bilevel DEA model takes into account stochastic conditions and optimizes centralized resource allocation and target setting, imposing lower bounds on the efficiencies of all DMUs affiliated to the organization. Consequently, the total input consumption is minimized, and the total output production is maximized while considering additional bounds and availability constraints for inputs. In the proposed bilevel model, uncertainty is introduced through the upper level (leader) problem that attempts to maximize organizational effectiveness, while in the lower level (follower) problem, the efficiency of the subordinate DMUs is evaluated. A solution methodology for the bilevel network DEA-based model is presented and numerical results are obtained using data from the literature. The obtained results are compared with those published in other case studies for centralized resource allocation DEA models.

1. Introduction

To increase their competitiveness and enhance their sustainability, decision-making units (DMUs) seek to allocate their resources optimally, maximize their output towards rational targeting and evaluate their efficiency systematically. At the same time, organizational resource allocation is of great importance, not only because resources are limited but also due to the fact that resource allocation has a serious impact on effectiveness, target setting and production planning. The process of resource distribution and target setting in organizations is usually implemented through a central unit that decides upon the resources supplied to the subordinate decision-making units (DMUs) along with DMUs lower bounds on desired efficiency. Moreover, the central unit has the authority to set the overall expected output targets so as to maximize the organizational effectiveness.

Data Envelopment Analysis (DEA) is a mathematical programming technique that has been extensively used for measuring the relative efficiency of homogeneous DMUs with multiple inputs and outputs [1,2]. Traditional DEA models can assess the efficiency of individual DMUs. However, such models are not suitable when one considers the scenario of multiple DMUs operating under the control of a higher-level supervising entity. In such a scenario, it is common that the higher-level entity seeks to maximize the efficiency of each DMU and at the same time minimize the total input consumption and/or maximize the total output production. This type of network structure is typical in organizations consisting of a central unit that makes centralized decisions affecting a number of subordinate decision-making units. The central unit decides for the allocation of limited resources to the DMUs and sets the output targets, which are influenced by the number of resources distributed to the DMUs. Typical organizations that exhibit this structure might include banks, hospitals, universities, etc.

DEA has been widely used for centralized resource allocation and various DEA-based resource allocation models have been published in the relevant literature. Beasley [3] presented DEA-based models for allocating fixed costs and input resources to DMUs as well as for target setting. Moreover, he reinterpreted DEA as a holistic selection of weights for all DMUs so as to maximize the average efficiency, while in traditional DEA, weights are chosen separately for each DMU. Lozano and Villa [4] presented two DEA models for centralized resource allocation, which minimize total input consumption or maximize total output production while considering the efficiency of the individual DMUs. Wu [5] presented a bilevel DEA model that optimizes the firm performance in decentralized companies. The model allocates resources between the two stages of a DMU, the (first) stage of the leader and the (second) stage of the follower, in a cost-efficient way. Hakim [6] proposed a bilevel DEA model for centralized resource allocation, where the organizational efficiency is maximized, satisfying at the same time, a lower bound on the efficiency of each DMU. Ang et al. [7] propose two-stage DEA models with bilevel formulations, where the upper-level maximizes the organizational effectiveness, while the lower-level model constrains the efficiency of all DMUs simultaneously. Furthermore, they consider two-stage DMUs, where the inputs of the first stage are converted into intermediate measures, which in turn are converted into outputs in the second stage.

In this paper, we evaluate organizational efficiency using a bilevel network DEA approach in a framework that introduces an aspect of uncertainty. The proposed model is based on the bilevel DEA model presented in Hakim et al. [6] and includes stochastic conditions. It attempts to optimize centralized resource allocation and target setting by imposing scenarios of lower bounds on the efficiencies of all DMUs belonging to the organization. Consequently, the total input consumption is minimized and, simultaneously, the total output production is maximized, while additional bounds and availability constraints (that is a stochasticity dimension) for inputs are considered. Concretely, in this bilevel DEA model, stochasticity takes the form of discrete scenarios associated with a user-defined occurrence probability. Each discrete scenario sets a value to the stochastic parameter of the model regarding input availability. Then, the expected total benefit for an organization is determined through a weighted average of the obtained optimal solutions based on the scenarios and their realization probabilities.

2. A Stochastic Bilevel DEA Model for Centralized Resource Allocation

Roughly speaking, a bilevel program is a mathematical programming problem whose feasible space encapsulates the parametric solution of another mathematical program. Moreover, such a structure consists of an upper-level optimization problem (leader’s problem) and a lower-level optimization problem (follower’s problem). Objective function, decision variables and parameters are defined for the upper and the lower-level problems, respectively. Bilevel programming has been used to model complex network structures of DMUs in a wide variety of fields, such as banking, engineering, supply chain management, ecology, etc. [8].

Conventional DEA is a popular method for the efficiency evaluation of DMUs, where all input and output data used are assumed to be accurate and deterministic. However, in many real-world problems, input and output data may be erroneous and unavailable due to information loss, human errors and a lack of historical data. For this reason, conventional DEA models have received a great deal of criticism, leading to a variety of extensions to negate this drawback (such as fuzzy logic, stochastic approaches, models for imprecise data, etc.) [9]. Apart from uncertainty in data, a solution may become infeasible or suboptimal in the implementation phase. Ben-Tal et al. [10] showed that even a small perturbation in data can lead to a considerable change in the feasibility of the optimal solution.

For the DMU efficiency evaluation with uncertain data, two approaches are the most common, namely robust and stochastic optimization. Soyster [11] sets the foundations for robust optimization by assigning each uncertain parameter in convex programming problems to its worst-case value within a set. Ben-Tal and Nemirovski ([10,12]) and El-Ghaoui and Lebret [13], who considered this approach to be too conservative, allowed the uncertain parameters to uncertainty sets without the most unlikely values and derived tractable mathematical programs. Bertsimas et al. ([14,15]) proposed a robust optimization approach where the constructed problems remain in the same class. Mulvey et al. [16] presented an approach that uses goal programming with a description of problem data based on scenarios. The number of studies that deal with stochasticity in the DEA framework is continuously growing. Omrani et al. [17,18] developed a multi-objective DEA model to determine three types of efficiency, i.e., profitability, operational and transactional for bank branches with uncertain data. The uncertainty in data is treated using discrete scenarios. The proposed models were tested on 45 Iranian Agriculture bank branches under four different scenarios. Shakouri et al. [19] present a p-robust DEA model to evaluate the efficiency of DMUs under uncertainty in data, where input parameters are given for different scenarios. Their model performed an efficiency assessment of the Iranian banking sector. Moreover, Shakouri et al. [20] presented network DEA models based on Stackelberg and game theory under uncertainty. They too applied their models on an analysis of bank branch performance.

In this paper, we propose a stochastic bilevel DEA-based model, which maximizes overall efficiency while considering, at the same time, a lower bound for the efficiency of the DMUs. In the proposed bilevel formulation, there are two submodels, the upper-level and the lower-level model. The upper-level determines the inputs and the outputs, which optimize overall efficiency by maximizing the total benefits (total outputs minus total inputs). The lower-level model computes the weights associated with the inputs and the outputs that maximize the efficiency of each subordinate DMU. Our approach exploits the leader–follower relationships in the bilevel framework that cannot be easily captured otherwise. To take into consideration the uncertainty aspect, we discretize the stochastic nature of resources upper bounds in the bilevel problem using an approach of different scenarios. Each discrete scenario sets a value to the stochastic parameter of the model in terms of input availability, along with a probability of realization that reflects the decision maker’s confidence or the aspiration of the specific scenario to be realized. Throughout the paper, we use the following notation.

Suppose we have $n$ DMUs, which have a bilevel network structure and each DMU $j$ $\left(j=1, \dots ,n\right)$ uses $m$ inputs $x_{ik}$ to produce $s$ outputs $y_{rk}$. Let $T=\left\{1,\dots ,N\right\}$ be a set of discrete scenarios, where each has an occurrence probability $w^t$. For each scenario $t\in T$, the upper-level optimization model for DMU $k$ $\left(k=1,2,\dots ,n\right)$ is the following:

$$\underset{x_{ik}^t,y_{rk}^t,{\lambda }_{jk}^t}{max}{\displaystyle \sum }_{r=1}^s{p}_r{\displaystyle \sum }_{k=1}^n{y}_{rk}^t-{\displaystyle \sum }_{i=1}^m{q}_i{\displaystyle \sum }_{k=1}^n{x}_{ik}^t$$

(1)

s.t.

$$L{e}_k\le e_k{}^{*} \left(k=1,2,\dots ,n\right)$$

(2)

$$x_{ik}^t\ge {\displaystyle \sum }_{j=1}^n{\lambda }_{jk}^t{X}_{ij} \left(i=1,2,\dots ,m;k=1,2,\dots , n\right)$$

(3)

$$y_{rk}^t\le {\displaystyle \sum }_{j=1}^n{\lambda }_{jk}^t{Y}_{rj} \left(r=1,2,\dots ,s;k=1,2,\dots , n\right)$$

(4)

$${\displaystyle \sum }_{j=1}^n{\lambda }_{jk}^t=1 \left(k=1,2,\dots ,n\right)$$

(5)

$${\lambda }_{jk}^t\ge 0 \left(j=1,2,\dots ,n;k=1,2,\dots , n\right)$$

(6)

$${\displaystyle \sum }_{k=1}^n{x}_{ik}^t\le b_i^t \left(i=1,2,\dots ,m\right)$$

(7)

$$L{x}_{ik}\le x_{ik}^t\le U{x}_{ik} \left(i=1,2,\dots ,m;k=1,2,\dots , n\right)$$

(8)

$$L{y}_{rk}\le y_{rk}^t\le U{y}_{rk} \left(r=1,2,\dots ,s;k=1,2,\dots , n\right)$$

(9)

In the objective Function (1) the unit costs $q_i$ of the inputs and the unit prices $p_r$ of the outputs are determined by the central unit. The optimal value of the objective function is denoted with $z^t$. Constraint (2) ensures that the efficiency of each subordinate DMU satisfies a lower bound set by the central unit. Constraints (3) and (4) ensure that the new input resources and the output targets belong to the production possibility set constructed by the observed inputs–outputs of all DMUs. The upper-level model is based on the variable returns to scale (VRS) assumption due to Constraint (5); however, it can also assume constant returns-to-scale (CRS), ignoring the latter constraint. Constraint (7) restricts the availabilities of resources with the stochastic parameter $b_i^t$. Constraints (8) and (9) set the lower and upper bounds for input resources and output targets, respectively. The lower-level optimization model for DMU $k$ $\left(k=1,2,\dots ,n\right)$ under scenario $t$ is the following:

$$e_k^{*}=\underset{v_{ik}^t,u_{rk}^t,l_k^t}{max}\frac{{{\displaystyle \sum }}_{r=1}^s{u}_{rk}^t{y}_{rk}^t-l_k^t}{{{\displaystyle \sum }}_{i=1}^m{v}_{ik}^t{x}_{ik}^t}$$

(10)

s.t.

$$0\le {\mathrm{e}}_{\mathrm{kj}}=\frac{{{\displaystyle \sum }}_{\mathrm{r}=1}^{\mathrm{s}}{\mathrm{u}}_{\mathrm{rk}}^{\mathrm{t}}{\mathrm{y}}_{\mathrm{rj}}^{\mathrm{t}}-{\mathrm{l}}_{\mathrm{k}}^{\mathrm{t}}}{{{\displaystyle \sum }}_{\mathrm{i}=1}^{\mathrm{m}}{\mathrm{v}}_{\mathrm{ik}}^{\mathrm{t}}{\mathrm{x}}_{\mathrm{ij}}^{\mathrm{t}}}\le 1 \left(j=1,2,\dots ,n\right)$$

(11)

$${\mathrm{u}}_{\mathrm{rk}}^{\mathrm{t}}\ge 0 \left(r=1,2,\dots ,s\right)$$

(12)

$${\mathrm{v}}_{\mathrm{ik}}^{\mathrm{t}}\ge 0 \left(i=1,2,\dots ,m\right)$$

(13)

The lower-level model is the standard DEA model, as presented in Beasley [3]. The objective Function (10) maximizes the efficiency of each DMU $k$. Constraint (11) restricts cross efficiency to take values between zero and one. The cross-efficiency $e_{kj}$ is defined as the efficiency of DMU $k$, when it is evaluated using the weights that are used to compute the efficiency of DMU $j$. Constraints (12) and (13) impose the nonnegativity of the output and input weights respectively. Due to the existence of the free variable $l_k^t$, the lower-level model computes the variable returns to scale efficiency of DMU $k$.

The stochastic bilevel DEA model is a non-linear programming problem, which cannot be solved in its bilevel form. Thus, the proposed bilevel DEA model is converted to a single level optimization problem, according to Theorem 1 in [6]. The single level problem for the proposed model is as follows:

$$\underset{x_{ik}^t,y_{rk}^t,v_{ik}^t,u_{rk}^t,l_k^t,{\lambda }_{jk}^t}{max}{\displaystyle \sum }_{r=1}^s{p}_r{\displaystyle \sum }_{k=1}^n{y}_{rk}^t-{\displaystyle \sum }_{i=1}^m{q}_i{\displaystyle \sum }_{k=1}^n{x}_{ik}^t$$

(14)

s.t.

$$L{e}_k\le e_{kk}=\frac{{{\displaystyle \sum }}_{r=1}^s{u}_{rk}^t{y}_{rk}^t-l_k^t}{{{\displaystyle \sum }}_{i=1}^m{v}_{ik}^t{x}_{ik}^t} \left(k=1,2,\dots ,n\right)$$

(15)

$${\mathrm{x}}_{\mathrm{ik}}^{\mathrm{t}}\ge {\displaystyle \sum }_{\mathrm{j}=1}^{\mathrm{n}}{\mathsf{\lambda }}_{\mathrm{jk}}^{\mathrm{t}}{\mathrm{X}}_{\mathrm{ij}} \left(i=1,2,\dots ,m;k=1,2,\dots , n\right)$$

(16)

$$y_{rk}^t\le {\displaystyle \sum }_{j=1}^n{\lambda }_{jk}^t{Y}_{rj} \left(r=1,2,\dots ,s;k=1,2,\dots , n\right)$$

(17)

$${\displaystyle \sum }_{j=1}^n{\lambda }_{jk}^t=1\left(k=1,2,\dots ,n\right)$$

(18)

$${\lambda }_{jk}^t\ge 0 \left(j=1,2,\dots ,n;k=1,2,\dots , n\right)$$

(19)

$${\displaystyle \sum }_{k=1}^n{x}_{ik}^t\le b_i^t \left(i=1,2,\dots ,m\right)$$

(20)

$$L{x}_{ik}\le x_{ik}^t\le U{x}_{ik} \left(i=1,2,\dots ,m;k=1,2,\dots , n\right)$$

(21)

$$L{y}_{rk}\le y_{rk}^t\le U{y}_{rk} \left(r=1,2,\dots ,s;k=1,2,\dots , n\right)$$

(22)

$$0\le e_{kj}=\frac{{{\displaystyle \sum }}_{r=1}^s{u}_{rk}^t{y}_{rj}^t-l_k^t}{{{\displaystyle \sum }}_{i=1}^m{v}_{ik}^t{x}_{ij}^t}\le 1 \left(k=1,2,\dots ,n;j=1,2,\dots , n\right)$$

(23)

$$u_{rk}^t\ge 0 \left(k=1,2,\dots ,n;r=1,2,\dots , s\right)$$

(24)

$$v_{ik}^t\ge 0 \left(k=1,2,\dots ,n;i=1,2,\dots , m\right)$$

(25)

The single-level problem is a non-linear/non-convex programming problem due to Constraints (15) and (23). The superscript $t$ in the variables of the single-level model is used mostly for notational convenience, since the model is separable over each scenario $t$. To estimate the optimal strategy for an organization concerning centralized resource allocation and target setting, we compute the expected total profit, which is a weighted average of the total optimized profits using as weights the occurrence probability for all scenarios.

3. Computational Results

In this section, we present preliminary computational results that we obtained by solving the proposed bilevel model for an example using data that appeared first in [21]. The single-level DEA-based model is implemented in Python with the use of the Pyomo library and solved on a PC with 16 GB RAM and CPU 2.6 GHz.

In this example, 10 DMUs are considered, which consume two inputs and produce two outputs. We considered three scenarios under uncertainty conditions, with the realization probabilities $w^1=0.3$, $w^2=0.2$ and $w^3=0.5$. For each scenario, we set the following upper bounds for the availability of the two inputs $b_1^1=98$, $b_2^1=90$, $b_1^2=100$, $b_2^2=110$, $b_1^3=90$ and $b_2^3=100$. The input and output costs and prices for the two inputs and the two outputs were $q_1=2$, $q_2=2$, $p_1=10$ and $p_2=5$. The upper and lower bounds for inputs and similarly for outputs were determined with the following rule: the upper input bound $U{x}_{ik}$ (upper output bound $U{y}_{rk}$) is 110% of the input $x_{ik}^t$ (output $y_{rk}^t$) and the lower input bound $L{x}_{ik}$ (lower output bound $L{y}_{rk}$) is the 90% of the input $x_{ik}^t$ (output $y_{rk}^t$). Furthermore, the lower-bound efficiencies for the ten DMUs are taken from Table 5 in [6]. The data we use for inputs and outputs are discretionary and non-categorical and are presented in Table 2 in [6]. The results obtained for each of the three scenarios are shown in

Table 1. Input and output results for the three scenarios.

DMUs	${\mathit{x}}_{1\mathit{k}}^1$	${\mathit{x}}_{2\mathit{k}}^1$	${\mathit{y}}_{1\mathit{k}}^1$	${\mathit{y}}_{2\mathit{k}}^1$	${\mathit{x}}_{1\mathit{k}}^2$	${\mathit{x}}_{2\mathit{k}}^2$	${\mathit{y}}_{1\mathit{k}}^2$	${\mathit{y}}_{2\mathit{k}}^2$	${\mathit{x}}_{1\mathit{k}}^3$	${\mathit{x}}_{2\mathit{k}}^3$	${\mathit{y}}_{1\mathit{k}}^3$	${\mathit{y}}_{2\mathit{k}}^3$
1	8.1	8.1	2.2	1.1	8.1	8.1	2.2	1.1	8.1	8.1	2.2	1.1
2	10.8	7.2	3.3	1.1	10.8	7.2	3.3	1.1	10.8	7.2	3.3	1.1
3	6.3	10.8	2.2	2.2	6.3	10.8	2.2	2.2	6.3	10.8	2.2	2.2
4	6.6	10.0	5.1	3.3	6.6	10.0	5.1	3.3	6.0	10.0	5.0	3.0
5	10.4	5.5	4.4	4.1	10.4	5.5	4.4	4.1	9.6	5.5	4.11	3.89
6	7.2	9.0	3.3	3.3	7.2	9.0	3.3	3.3	7.2	9.0	3.3	3.29
7	12.3	9.4	6.3	5.4	12.3	9.4	6.3	5.4	10.8	10.0	5.8	5.4
8	14.0	6.0	8.0	2.0	14.0	6.0	8.0	2.0	12.7	6.6	7.45	2.19
9	10.8	10.8	1.1	5.7	10.8	10.8	1.1	5.7	10.8	10.8	1.1	5.7
10	8.4	8.2	3.3	5.1	8.4	8.2	3.3	5.1	7.7	8.3	3.3	4.7

. These results describe the optimal allocation of resources and output targets achievement for an organization with 10 subordinate DMUs in each scenario. The profits are $198.533$ for scenario 1 and 2 and $188.046$ for scenario 3. The expected total profit is $193.29$.

4. Conclusions

The stochastic bilevel DEA model that we presented considers leader–follower relations, the desirable efficiency for DMUs and data uncertainty at the same time. One of the main advantages of this model is that it enables decision makers to obtain an optimal strategy for resource allocation and output targeting in a holistic manner, taking into consideration the efficiency objectives of DMUs. To overcome the difficulties in estimating the future optimal organizational benefits when data uncertainty is taken into account, we rely on an approach that considers discrete scenarios for each of the stochastic model parameters in order to determine the total expected profit. The expected profit of each scenario is realized using discrete probability distribution determining the occurrence event of each scenario. Of course, limitations exist, and several issues need to be further investigated. The number of scenarios that must be designed ex ante and the computational needs of the problem for a large number of scenarios are under investigation. Another issue is the development of a two-stage stochastic model with recourse action and emphasis on observed inputs/outputs when it comes to uncertainty. Finally, additional work is needed in the process of actually specifying the scenarios and in improving the existing solution methodologies for the stochastic bilevel DEA model.