A comprehensive method for the centralized resource allocation in DEA

Accepted Manuscript A Comprehensive Method for the Centralized Resource Allocation in DEA. J. Sadeghi, A. Dehnokhalaji PII: DOI: Reference:

S0360-8352(18)30481-9 https://doi.org/10.1016/j.cie.2018.10.011 CAIE 5450

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Computers & Industrial Engineering

Received Date: Revised Date: Accepted Date:

28 February 2018 30 September 2018 5 October 2018

Please cite this article as: Sadeghi, J., Dehnokhalaji, A., A Comprehensive Method for the Centralized Resource Allocation in DEA., Computers & Industrial Engineering (2018), doi: https://doi.org/10.1016/j.cie.2018.10.011

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*Title Page including Author Details

A Comprehensive Method for the Centralized Resource Allocation in DEA J. Sadeghi a, A. Dehnokhalaji b a

Department of Mathematics, Faculty of Mathematical Sciences and Computer, Kharazmi University, 50, Taleghani Ave., 1561836314, Tehran, Iran. b Department of Computer Science, Faculty of Mathematical Sciences and Computer, Kharazmi University, 50, Taleghani Ave., 1561836314, Tehran, Iran.

E-mail address, Telephone and Fax: Author name: Jafar Sadeghi (Corresponding author) E-mail: [email protected] ; [email protected] Tel.: +982177630040 Fax: +982177696359

Author name: Akram Dehnokhalaji E-mail: [email protected] Tel.: +982177630040

A Comprehensive Method for the Centralized Resource Allocation in DEA.

February 28, 2018 First revision: September 3, 2018 This version: September 30, 2018

Abstract In this paper, we propose two new centralized resource allocation methods extending Lozano and Villa’s method (2004) to more general cases. The main assumption in this study is that all Decision Making Units (DMUs) are operating under the supervision of a central DMU in order to introduce the input and output targets for all units in the next production period. This paper considers two planning ideas. The first one is maximizing outputs produced with future planned resources and removing all input inefficiencies as much as possible while making all units strongly efficient. The second one is optimizing total revenue and cost functions to reach the best performance of the system. All units are assumed to be able to adjust their input consumptions and output productions in the current Production Possibility Set (PPS). Finally, we show that the two proposed methods provide the same results when input and output prices are replaced with their corresponding shadow prices. The proposed methods are illustrated with two empirical data sets.

Keywords: Data Envelopment Analysis; Resource Allocation; Cost Function; Revenue Function.

1

1

Introduction

Data Envelopment Analysis (DEA) is an effective approach for evaluating the performance of homogeneous Decision Making Units (DMUs), first proposed by Charnes et al. (1978). In recent years, several studies have been developed regarding the applications of DEA in educational institutes, industries, banks, etc. (For some of the applications using DEA, see: Kerstens et al. (2006); Sahoo and Tone (2009); Toloo et al. (2018)). One important application of DEA is resource allocation. Most of the studies about the resource allocation in DEA literature can be generally divided into two categories. The first category includes the papers which focus on allocating fixed costs to DMUs. Fixed cost is a total fixed (overhead) cost that is used for the common infrastructures for the subunits of an organization. This problem is a common one encountered in organizational budgeting/costing, namely to split an overhead cost amongst different departments (Beasley (2003)). Basically, Cook and Kress (1999) and Beasley (2003) made the first effort to solve the allocating fixed cost using DEA. There is a large volume of published studies describing the role of allocating fixed costs (e.g. Li et al. (2009); Si et al. (2013); Li et al. (2013);Lozano (2014); Li et al. (2017); Jahanshahloo et al. (2017); Li et al. (2018a); Li et al. (2018b)). The second category includes papers that give an account of allocating input resources (such as money and manpower) to DMUs. Simultaneous to allocating input resources output targets are also decided for each DMU. In this paper, we only address the second category i.e., resource allocation and target setting among DMUs where the Decision Maker (DM) allocates available resources to units in order to achieve a specific goal. DEA methods make a new viewpoint in resource allocation problem as it is possible to obtain feasible input and output targets by trade-offs among inputs and outputs based on empirical properties of a PPS ( Korhonen and Syrj¨anen (2004); Amirteimoori and Emrouznejad (2012); Silva and Milioni (2012); Fang (2016)). The traditional resource allocation models in DEA evaluate each unit individually. However, in most applications, there are conditions in which a centralized DM superintends all units, and (s)he can control some managerial situation, for instance, allocating available resources to units. This type of resource allocation is called the centralized resource allocation problem by Lozano and Villa (2004). The idea behind their method is to decrease the total input or increase the total output of all units simultaneously. Asmild et al. (2009) presented a modified form of one of the centralized models proposed by Lozano and Villa (2004) only to consider the adjustment of previous inefficient units. Mar-Molinero et al. (2014) utilized

2

Lozano and Villa’s model (2004) and simplified it to implement in many conditions and to identify the most efficient unit. Fang (2013) developed a new generalized centralized resource allocation model based on models proposed by Lozano and Villa (2004) and Asmild et al. (2009). Fang (2015) suggested a centralized approach which obtains a sequence of intermediate benchmarks to improve the efficiency value. There are also a number of studies in DEA literature that focus on centralized resource allocation (e.g. Golany et al. (1993); Golany and Tamir (1995); Athanassopoulos (1995); Beasley (2003); Lozano et al. (2004); Lozano and Villa (2005); Pachkova (2009); Lozano et al. (2009); Du et al. (2010); Lozano et al. (2011); Lotfi et al. (2013); Fang and Li (2015); Fang (2016); Dehnokhalaji et al. (2017)). However, most researchers in DEA literature have addressed centralized resource allocation models from the reallocation perspective i.e., allocating the current resources to units as well. However, in some organizations, there is a situation in which the amount of the current resources should be increased (i.e., allocating additional amount of resources among units) or reduced (i.e., allocating less amount of resources among units), in the next production period. It seems that, to date, the extra or shortage resource-allocation problem has not been addressed sufficiently in literature. For example, Beasley (2003), Korhonen and Syrj¨anen (2004) and Nasrabadi et al. (2012) have addressed these issues. This kind of extra or shortage resource-allocation problem can frequently be observed in practice. For example, the top manager of bank branches aims to assign a large amount of premium to their branches; a company aims to allocate some bonus to selected staff members at the end of the year; a factory needs to reduce labor force because of being on a tight budget. This study aims to answer the following question: How should we distribute the premium, bonus or allocate less labor among units to achieve the fair principle and meanwhile make it beneficial to all? In this paper, we extend the centralized resource allocation method by Lozano and Villa (2004) to allocate planned future resources across a set of DMUs. In the proposed models one can consider both variable and constant returns to scale cases. Also, the results of allocation, i.e., input and output targets, belong to the original PPS. Compared with the other methods, our approach has the following scientific advantages: (i) Lozano and Villa (2004) presented the centralized resource allocation models that provide the reallocation of current inputs, but they did not consider increased or decreased of resources in the next period. This paper includes these extra situations and their corresponding managerial interpretations. We then extend the model to incorporate non-discretionary variables and obtain more attainable 3

targets incorporating managerial limitations. One of key features of the proposed model is the fact that it projects all DMUs onto the efficient frontier simultaneously. (ii) This paper also develops another centralized resource allocation model to improve the total amount of revenue and reduce the total amount of cost by allocating the further resources among all units. We also show that the initial proposed centralized DEA model is a special case of our second model when the prices of inputs and outputs are replaced with their shadow prices. The rest of this paper is organized as follows: Section 2 provides some background on DEA and reviews the non-radial version of the centralized resource allocation model proposed by Lozano and Villa (2004). Section 3 develops the centralized DEA models proposed by Lozano and Villa (2004) in such a way that all kinds of allocations including additional, current and shortage resource allocation cases are covered. Section 4 formulates a centralized resource allocation model based on optimizing the total revenue and cost functions. In Section 5, we compare our method with some existing methods and apply our approach to analyzes two empirical examples. Section 6 presents the concluding remarks.

2

Technology: Basic Preliminaries

Consider n homogeneous decision-making units DMUj , j = 1, ..., n with input values xij and outputs yrj where i = 1, ..., m, r = 1, ..., s. Let all units be under the supervision of a centralized Decision Maker (DM). This situation happens when all units belong to the same organization. The famous Production Possibility Set (PPS) in DEA is denoted as: T =

(

(x, y) |

n X

λj xj ≤ x,

j=1

n X j=1

)

λj yj ≥ y, λ ∈ Γ ,

where Γ = Rn+ denotes the PPS with the constant returns to scale (CRS) assumption, and Pn Γ = {λ ∈ Rn+ | j=1 λj = 1} denotes the PPS with the variable returns to scale (VRS) assumption.

Definition 2.1. A point (¯ x, y¯) ∈ T is called strongly efficient (or non-dominated) if and only if there does not exist another (x, y) ∈ T such that x ≤ x¯, y ≥ y¯ and (x, y) 6= (¯ x, y¯).

4

2.1

Lozano and Villa (2004)’s approach

For resource reallocation, Lozano and Villa (2004) presented the centralized resource allocation models. They only focus on radial and non-radial input-oriented models. Here, however, we consider the non-radial output-oriented case. In principle, using a non-radical model appears more suitable than the traditional radial model because the latter would not measure all input and output inefficiencies. The first phase of the output-oriented for the centralized resource allocation can be formulated as follows: s P

z ∗ = max

ϕr r=1 n P n P

s.t

k=1 j=1 n P n P

λkj xij ≤

n P

xik ,

k=1

λkj yrj ≥ ϕr

k=1 j=1

n P

i = 1, ..., m,

yrk , r = 1, ..., s,

(1)

k=1

ϕr ≥ 1,

r = 1, · · · , s,

λk ∈ Γ, where j, k = 1, ..., n indicate the indices of DMUs. The phase II model can be formulated as: max s.t

m P

i=1 n P

si n P

k=1 j=1 n P n P

λkj xij =

n P

xik − si , i = 1, ..., m,

k=1

λkj yrj = ϕ∗r

k=1 j=1 k

n P

yrk ,

(2)

r = 1, ..., s,

k=1

λ ∈ Γ,

where ϕ∗r is the optimum value of ϕr in the model (1). After solving model (2), the input and output target for DMUk , k = 1, ..., n, can be defined as n P x¯ik = λk∗ j xij , i = 1, ..., m, y¯rk =

j=1 n P

λk∗ j yrj , r = 1, ..., s,

j=1

k∗ where vector (λk∗ 1 , ..., λn ) is an optimal solution of model (2). Lozano and Villa (2004)

provided that all target units are strongly efficient simultaneously.

5

3

Model development

In models (1) and (2), Lozano and Villa (2004) reallocated the current resources to a set of homogeneous units. However, there is a situation in which the current resources should be increased or reduced. In this section, we extend their central method and propose a model to allocate additional or less resources to a set of homogeneous units operating in a centralized decision-making environment. Before formulating appropriate models, model (1) can be written as follows (see: Theorem 3.1): z1∗ = max s.t

s P

r=1 n P

k=1 n P

ϕr x¯ik ≤

n P

y¯rk ≥ ϕr

i = 1, ..., m,

(3.1)

yrk , r = 1, ..., s,

(3.2)

xik ,

k=1

n P

k=1

k=1

ϕr ≥ 1, n P λkj xij ≤ x¯ik , j=1 n P

r = 1, · · · , s,

(3.3)

(3)

i = 1, ..., m, k = 1, ..., n, (3.4)

λkj yrj ≥ y¯rk ,

j=1 k

λ ∈ Γ, x¯ij ≥ 0, y¯rj ≥ 0,

r = 1, ..., s, k = 1, ..., n,

(3.5)

k = 1, ..., n,

(3.6) (3.7)

where j, k = 1, ..., n indicate the indices of DMUs, and x¯ik and y¯rk are the targets for i-th input and r-th output of DMUk , respectively. Constraint (3.1) makes trade-offs among inputs components, namely if the input of some units increased (decreased) then some other units decrease (increase) their inputs, such that the total consumption of different inputs remains below the current total level. Constraints (3.2) and (3.3) show that for every output, the total target of the system is the sum of the individual targets and denotes a certain increase percentage with respect to the current total output level. Constraints (3.4), (3.5) and (3.6) indicate that units are allowed to change their resources only within the current PPS and non-negativity of decision variables are ensured by (3.7). The following theorem shows that model (1) and model (3) provide the same results. Theorem 3.1. The production plans obtained by solving models (1) and (3) are identical. ∗ Proof. Suppose that (λk∗ ¯∗ik , y¯rk , ϕ∗r ) is an optimal solution of model (3). By combining j ,x

6

constraints (3.1), (3.2), (3.4) and (3.5), we have: n P n P

k=1 j=1 n P n P

n P

∗

λkj xij ≤

λkj yrj ≥ ϕ∗r ∗

k=1 j=1 k

i = 1, ..., m,

xik ,

k=1

n P

yrk , r = 1, ..., s,

(4)

k=1

λ ∈ Γ,

k = 1, ..., n.

Therefore, (λkj , ϕ∗r ) is a feasible solution of model (1), hence, we have z ∗ ≥ z1∗ . On the other ¯ k , ϕ¯r ) is an optimal solution of model (1), hence, by letting hand, suppose that (λ ∗

j

x¯∗ik = ∗ y¯rk

=

n P ¯ k xij , λ j

j=1 n P

j=1

¯ k yrj , λ j

(5)

and considering constraints of model (1), we have: n P

k=1 n P

x¯∗ik ≤

n P

xik ,

k=1 ∗ y¯rk

≥ ϕ¯r

k=1

n P

i = 1, ..., m, (6)

yrk , r = 1, ..., s,

k=1

Also, based on equation (5), we have: n P ¯ k xik ≤ x¯∗ , i = 1, ..., m, λ j ik

j=1 n P

j=1 k

¯ k yrk ≥ y¯∗ , r = 1, ..., s, λ j rk

λ ∈ Γ,

(7)

k = 1, ..., n.

¯ k , x¯∗ , y¯∗ , ϕ¯r ) is a feasible solution for model (3) and z ∗ ≤ z ∗ . Hence, we have Therefore, (λ j 1 ik rk z ∗ = z1∗ . Note that, based on (5), both models (1) and (3), provide the same targets. As it was mentioned before, in some organizations the policy of production is increasing or decreasing current overall resources in the next production period. Thus, we propose the

7

following centralized resource allocation model: max s.t

s P

r=1 n P

k=1 n P

ϕr x¯ik ≤ αi y¯rk ≥ ϕr

k=1

n P

k=1 n P

xik , i = 1, ..., m, yrk , r = 1, ..., s,

k=1

ϕr ≥ 1, n P λkj xij ≤ x¯ik , j=1 n P

r = 1, · · · , s,

(8)

i = 1, · · · , m, k = 1, ..., n,

λkj yrj ≥ y¯rk ,

r = 1, ..., s, k = 1, ..., n,

j=1 k

λ ∈ Γ,

k = 1, ..., n,

x¯ij ≥ 0, y¯rj ≥ 0. The only difference between models (3) and (8) is the first constraint where parameter αi presents the amount of the allocated future resource i. In the case that the current resource should be reallocated, αi = 1. If the centralized DM wishes to increase the amount of i-th input, we have αi > 1. If the centralized DM plans to decrease the amount of i-th resource, we have αi < 1. Notice that, when at least one of parameters αi , i = 1, · · · , m, is below one it may be possible that the constraints ϕr ≥ 1, r = 1, · · · , s, may lead to infeasibility. Of course, this never happens if αi ≥ 1, i = 1, · · · , m. However, in this case, when we have infeasibility problem, it means that by decreasing some inputs components, all output components can not be increased. Ignoring the constraints ϕr ≥ 1, r = 1, · · · , s, leads to possible reductions (which can be large) in some outputs. To solve this problem one can choose a suitable weight wr > 0, r = 1, · · · , s, to allow those reductions and control them (Lozano and Villa (2004)). P Therefore, we replace the objective function of model (8) by sr=1 wr ϕr . Model (8) removes the outputs inefficiencies, but in order to remove input inefficiencies

and obtain strongly efficient targets, Lozano and Villa (2004) proposed the second phase model (2) that minimizes the sum of input slacks. Since the first phase model is formulated based on non-radial output measure, we formulate a second phase model also based on nonradial input measure. Hence, our second phase model minimizing the average percentage decrease in total input for the optimal solution of first phase model can be formulated as

8

follows: z2∗ = min s.t

m P

i=1 n P

k=1 n P

k=1 n P j=1 n P

θi x¯ik ≤ θi αi y¯rk ≥ ϕ∗r

n P

xik , i = 1, ..., m,

k=1 n P

yrk ,

r = 1, ..., s,

k=1

λkj xij ≤ x¯ik ,

i = 1, ..., m, k = 1, ..., n,

λkj yrj ≥ y¯rk ,

r = 1, ..., s, k = 1, ..., n,

j=1 k

λ ∈ Γ,

(9)

k = 1, ..., n,

x¯ij ≥ 0, y¯rj ≥ 0, where ϕ∗r is the optimal solution of model (8). As explained above, model (8) maximizes outputs produced with the future planned resources and model (9) removes all input inefficiencies as much as possible while making all units strongly efficient. Therefore, the target point (¯ xj , y¯j ) for DMUj , j = 1, ..., n, obtained by solving model (9), is strongly efficient. Remark 1: It is important to note that some certain inputs and outputs are not controllable by the central DM and are determined exogenously with no discretion from the central DM. We refer to this situation as strictly non-discretionary inputs and outputs. Previous researchers have considered this situation, typically by designating these factors as nondiscretionary (e.g. Charnes et al. (1984); Banker and Morey (1986)). Therefore, the subsets I D , I N D ⊆ {1, ..., m} denotes the index sets for discretionary and non-discretionary inputs, respectively, with the property that I D ∪ I N D = {1, ..., m}. In a similar way, the subsets O D , O N D ⊆ {1, ..., s} denote the index sets for discretionary and nondiscretionary outputs, respectively, with the property that O D ∪ O N D = {1, ..., s}. Remark 2: We suppose that (re)allocation does not change the current PPS. Furthermore, we assume that the units can modify their production in the defined PPS only according to certain limits that represent factors (managerial, environmental, etc.) that limit possible changes during the planning period (Korhonen and Syrj¨anen (2004)). We consider the following managerial constraints to set T which is called the managerial possibility set. i ∈ I D , k = 1, ..., n,

(10)

r ∈ O D , k = 1, ..., n,

(11)

x¯ik ≤ Ui xik y¯rk ≥ yrk

9

where Ui is an upper bound for i-th input. In fact, I D represents the subset of discretionary inputs which can be reallocated among units and x¯ik and y¯rk are the targets for i-th input and r-th output of DMUk for the next period, respectively. These managerial constraints insure the attainability of input targets and it is supposed to be valid in the neighborhood of the units’ current production. Also, Constraints (11) shows that the expected values for outputs should be more than their current level. The proposed approach can be easily extended in different directions. For example, the consideration of lower bounds on individual inputs or outputs or on the inequality of the allocated resources, etc., such as Li xik ≤ x¯ik ≤ Ui xik i ∈ I D , k = 1, ..., n, but then one must be aware that such constraints can prevent the projection of all DMUs onto the efficiency frontier which is a most desirable feature of the proposed models. According to Remarks 1 and 2, models (8) can be rewritten as follows: max s.t

P

w r ϕr

r∈O D n P

x¯ik ≤ αi

k=1 n P

k=1 n P j=1 n P

j=1 n P

j=1 n P

j=1

y¯rk ≥ ϕr

n P

k=1 n P

xik , i ∈ I D , yrk , r ∈ O D ,

k=1

λkj xij

i ∈ I D , k = 1, ..., n,

≤ x¯ik ,

i ∈ I N D , k = 1, ..., n,

λkj xij ≤ xik , λkj yrj

(12) D

≥ y¯rk ,

r ∈ O , k = 1, ..., n,

λkj yrj ≥ yrk ,

r ∈ O N D , k = 1, ..., n,

x¯ik ≤ Ui xik ,

i ∈ I D , k = 1, ..., n,

y¯rk ≥ yrk ,

r ∈ O D , k = 1, ..., n,

λk ∈ Γ,

k = 1, ..., n,

x¯ij ≥ 0, y¯rj ≥ 0. In cases that constraints (11) are imposed the inequalities ϕr ≥ 1, r ∈ O D , are not needed because they are redundant. Hence, we replace ϕr ≥ 1, r ∈ O D , by constraints (11). Note that, constraints (11) are even more constraining than ϕr ≥ 1, r ∈ O D , and hence if they are imposed, infeasibility may occur again. To solve this problem we replace the constraints y¯rk ≥ yrk , r ∈ O D , k = 1, ..., n by

10

y¯rk ≥ δ ∗ yrk , r ∈ O D , k = 1, ..., n, where δ ∗ is the optimal value of the following model. δ ∗ = max s.t

min {δr } n n P P xik , x¯ik ≤ αi

r∈O D

k=1 n P

k=1 n P

y¯rk ≥ ϕr

k=1 n P

yrk ,

i ∈ I D, r ∈ OD ,

k=1

λkj xij ≤ x¯ik ,

i ∈ I D , k = 1, ..., n,

j=1 n P

λkj xij ≤ xik ,

i ∈ I N D , k = 1, ..., n,

j=1 n P

λkj yrj ≥ yrk ,

j=1 n P

(13) λkj yrj

≥ y¯rk ,

D

r ∈ O , k = 1, ..., n, r ∈ O N D , k = 1, ..., n,

j=1

x¯ik ≤ Ui xik ,

i ∈ I D , k = 1, ..., n,

y¯rk ≥ δr yrk ,

r ∈ O D , k = 1, ..., n,

λk ∈ Γ,

k = 1, ..., n,

x¯ij ≥ 0, y¯rj ≥ 0, δr ≥ 0. The objective function of model (13) maximizes the minimal value of δr , r ∈ O D . The optimal solution of model (13) is not unique. Such solutions are called efficient solutions in MultipleCriteria Decision Making (MCDM) literature. In such circumstances, each efficient solution is acceptable (Korhonen and Syrj¨anen (2004)). We are able to transform model (13) into a

11

single-objective linear model by introducing the parameter η as follows: δ ∗ = max η s.t

η ≤ δr , r ∈ OD , n n P P xik , i ∈ I D , x¯ik ≤ αi k=1 n P

k=1 n P

j=1 n P

j=1 n P j=1 n P

y¯rk ≥ ϕr

k=1 n P

yrk , r ∈ O D ,

k=1

λkj xij

≤ x¯ik ,

i ∈ I D , k = 1, ..., n,

λkj xij ≤ xik ,

i ∈ I N D , k = 1, ..., n,

λkj yrj ≥ y¯rk ,

r ∈ O D , k = 1, ..., n,

λkj yrj ≥ yrk ,

r ∈ O N D , k = 1, ..., n,

(14)

j=1

x¯ik ≤ Ui xik ,

i ∈ I D , k = 1, ..., n,

y¯rk ≥ δr yrk ,

r ∈ O D , k = 1, ..., n,

λk ∈ Γ,

k = 1, ..., n,

x¯ij ≥ 0, y¯rj ≥ 0. Note that, if αi ≥ 1, i ∈ I D , then, we have δ ∗ ≥ 1. If αi ≤ 1 for some i ∈ I D , then, it may happen that δ ∗ < 1. To sum up the proposed method in this section, phase I and II can be rewritten as follows:

12

The phase I model which improves the total output is: max s.t

P

w r ϕr

r∈O D n P

x¯ik ≤ αi

k=1 n P

k=1 n P j=1 n P

j=1 n P

j=1 n P

y¯rk ≥ ϕr

n P

xik , i ∈ I D ,

k=1 n P

yrk , r ∈ O D ,

k=1

λkj xij

i ∈ I D , k = 1, ..., n,

≤ x¯ik ,

λkj xij ≤ xik , λkj yrj

i ∈ I N D , k = 1, ..., n, (15) D

≥ y¯rk ,

r ∈ O , k = 1, ..., n,

λkj yrj ≥ yrk ,

r ∈ O N D , k = 1, ..., n,

j=1

x¯ik ≤ Ui xik ,

i ∈ I D , k = 1, ..., n,

y¯rk ≥ δ ∗ yrk ,

r ∈ O D , k = 1, ..., n,

λk ∈ Γ,

k = 1, ..., n,

x¯ij ≥ 0, y¯rj ≥ 0. The phase II model which reduces the total input is: min s.t

P

θi

i∈I D n P k=1 n P

k=1 n P

j=1 n P

j=1 n P

j=1 n P

x¯ik ≤ θi αi y¯rk ≥

n P

xik , i ∈ I D ,

k=1

ϕ∗r

n P

yrk ,

r ∈ OD ,

k=1

λkj xij

i ∈ I D , k = 1, ..., n,

≤ x¯ik ,

λkj xij ≤ xik , λkj yrj

i ∈ I N D , k = 1, ..., n, (16) D

≥ y¯rk ,

r ∈ O , k = 1, ..., n,

λkj yrj ≥ yrk ,

r ∈ O N D , k = 1, ..., n,

j=1

x¯ik ≤ Ui xik ,

i ∈ I D , k = 1, ..., n,

y¯rk ≥ δ ∗ yrk ,

r ∈ O D , k = 1, ..., n,

λk ∈ Γ,

k = 1, ..., n,

x¯ij ≥ 0, y¯rj ≥ 0, where ϕ∗r is the optimal solution of model (15) and δ ∗ is the optimal value of the models 13

(14). In this section, we extended Lozano and Villa (2004) model and formulated models (15) and (16) by maximizing the total production and minimizing the total consumption. However, when input and output prices are known, we present a centralized DEA model to allocate (or reallocate) resources based on optimizing the revenue and cost functions across a set of DMUs. Hence, we extend models (15) and (16) and show the relationship of the resulting model with it in next section.

4

Resource allocation model based on optimizing the revenue and cost functions

As mentioned above, we assume that all units operate under the supervision of a central unit which can control some managerial parameters, such as resources. In such environment, the central DM aims to allocate (or reallocate) future resources among all observed units to optimize the total revenue and cost efficiency of the central unit. Thus, we formulate the central resource allocation model based on improving the total revenue and cost in two phases I and II as follows:

14

The phase I model which improves the total revenue is: max s.t

n P P

pr y¯rj

j=1 r∈O D n P

x¯ik ≤ αi

k=1 n P

k=1 n P j=1 n P

j=1 n P

j=1 n P

j=1

y¯rk ≥ ϕr

n P

k=1 n P

xik , i ∈ I D , yrk , r ∈ O D ,

k=1

λkj xij

i ∈ I D , k = 1, ..., n,

≤ x¯ik ,

λkj xij ≤ xik ,

i ∈ I N D , k = 1, ..., n,

λkj yrj ≥ y¯rk ,

r ∈ O D , k = 1, ..., n,

λkj yrj ≥ yrk ,

r ∈ O N D , k = 1, ..., n,

x¯ik ≤ Ui xik ,

i ∈ I D , k = 1, ..., n,

y¯rk ≥ δ ∗ yrk ,

r ∈ O D , k = 1, ..., n,

λk ∈ Γ,

k = 1, ..., n,

(17)

x¯ij ≥ 0, y¯rj ≥ 0. The phase II model which reduces the total cost is: max s.t

n P P

ci x¯ij

j=1 i∈I D n P

x¯ik ≤ αi

k=1 n P

k=1 n P j=1 n P

j=1 n P

j=1 n P

y¯rk ≥ ϕr

n P

k=1 n P

xik , i ∈ I D , yrk , r ∈ O D ,

k=1

λkj xij

i ∈ I D , k = 1, ..., n,

≤ x¯ik ,

λkj xij ≤ xik ,

i ∈ I N D , k = 1, ..., n,

λkj yrj ≥ y¯rk ,

r ∈ O D , k = 1, ..., n,

λkj yrj ≥ yrk ,

r ∈ O N D , k = 1, ..., n,

j=1

x¯ik ≤ Ui xik ,

i ∈ I D , k = 1, ..., n,

y¯rk ≥ δ ∗ yrk ,

r ∈ O D , k = 1, ..., n,

λk ∈ Γ,

k = 1, ..., n,

x¯ij ≥ 0, y¯rj ≥ 0 where ci and pr are the prices of i-th input and r-th output, respectively. 15

(18)

The objective function is the main difference between two phases models (15) and (16) and two phases models (17) and (18). Models (15) and (16) optimizes the total production and the total consumption, respectively, while models (17) and (18) optimizes the total revenue and cost functions, respectively. Fang (2016) proposed the centralized DEA model based on improving the total revenue efficiency. In the extreme case that the output price is not available, he proposed using absolute shadow prices. Based on this idea, we investigate the extreme case when no information about input costs or output prices are available. Hence, the following theorem shows the relation between the two proposed methods. Theorem 4.1. Assume that input and output prices are replaced with the shadow prices corresponding to variables θi and ϕr of model (15), respectively. Then targets for any DMUo obtained by solving models (16) and (18) are identical. Proof. Let wr = 1, r ∈ O D . First, we assume that the dual variables p˜r , r ∈ O D are the shadow prices corresponding to variables ϕr , r ∈ O D , in model (15). P From the dual of model (15), we get: p˜r nk=1 yrk = 1. Therefore, we have: p˜r =

Pn 1

P

ϕr =

P

P

ϕr =

k=1 yrk

(19)

.

Also, based on model (15) we have:

r∈O D

r∈O D

Pn ¯rk k=1 y Pn . k=1 yrk

Now from equation (19), we have,

r∈O D

n P P

p˜r y¯rk .

r∈O D k=1

Similarly, assume that dual variables c˜i , i ∈ I D are shadow prices corresponding to variP ables θi , i ∈ I D , in model (16). We have: c˜i αi nk=1 xik = 1. Hence, c˜i = αi Pn1 xik and k=1 P Pn ˜i xik = 1. Therefore, from model (16), we conclude that: i∈I D k=1 c P

i∈I D

θi =

P

i∈I D

Pn ¯ik k=1 x P αi n k=1 xik

16

=

n P P

i∈I D k=1

c˜i x¯ik .

Theorem 4.1 states that in the special case that the input and output prices in models (17) and (18) are the optimal shadow prices of models (15) and (16), both methods are equivalent.

5

Illustrative examples

In this section we illustrate our proposed approach through two empirical examples under the VRS assumption. In the first example, we assume that the input and output prices are not available. We use the data for 25 supermarkets in Finland taken from Korhonen and Syrj¨anen (2004). The supermarkets belong to the same chain store and under the supervision of the central DM that can control their performance and can allocate resources to them. Thus, the central DM can simultaneously manage these supermarkets to maximize the total output consumption by all units. Hence, it leads to the use of the centralized resource allocation models (15) and (16). Note that we assumed that w1 = w2 = 1. The information about inputs (Man-hours and Size) and outputs (Sales and Profit) are reported in Table 1. Man-hours refers to the labor used within a certain period and Size is the total retail floor space of the supermarket. We also assume that the supermarkets cannot adjust their floor space, but labors are assumed adjustable. Also, both outputs are discretionary variables.

17

Table 1: Data set for 25 supermarkets DMUj x1j x2j y1j y2j 1 79.1 4.99 115.3 1.71 2 60.1 3.3 75.2 1.81 3 126.7 8.12 225.5 10.39 4 153.9 6.7 185.6 10.42 5 65.7 4.74 84.5 2.36 6 76.8 4.08 103.3 4.35 7 50.2 2.53 78.8 0.16 8 44.8 2.47 59.3 1.3 9 48.1 2.32 65.7 1.49 10 89.7 4.91 163.2 6.26 11 56.9 2.24 70.7 2.8 12 112.6 5.42 142.6 2.75 13 106.9 6.28 127.8 2.7 14 54.9 3.14 62.4 1.42 15 48.8 4.43 55.2 1.38 16 59.2 3.98 95.9 0.74 17 74.5 5.32 121.6 3.06 18 94.6 3.69 107 2.98 19 47 3 65.4 0.62 20 54.6 3.87 71 0.01 21 90.1 3.31 81.2 5.12 22 95.2 4.25 128.3 3.89 23 80.1 3.79 135 4.73 24 68.7 2.99 98.9 1.86 25 62.3 3.1 66.7 7.41 Total 1901.5 102.97 2586.1 81.72 x1j : Man-hours (103 h); x2j : Size (103 m2 ) y1j : Sale (106 FIM); y2j : Profit (106 FIM) We consider three different cases for resource allocation, in all cases, we assume that the units are able to modify the amount of their inputs and outputs proportionally in the neighborhood of their current production. In the first case, the DM’s policy is decreasing the total Man-hours for all supermarkets by 15%, hence α1 = 0.85. To ensure the managerial feasibility, the change of units input is limited to a 5% increasing, so U1 = 1.05. In the second case, the DM’s policy is decreasing the total Man-hours for all supermarkets by 5%, hence α1 = 0.95. To ensure the managerial feasibility, the change of units input is 18

limited to a 10% increasing, so U1 = 1.1. In the third case, the total amount of the first input of all units can be increased by at most 10%(α1 = 1.1) and to guarantee managerial feasibility, the change in inputs of each unit is limited to 20% increase (U1 = 1.2). Table 2: Results for 25 supermarkets DMUj 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

x¯1j 64.96 50.02 121.70 130.52 54.01 64.31 49.26 44.80 48.10 86.82 56.90 77.18 70.55 45.07 44.86 56.27 67.86 62.68 45.71 47.19 59.51 71.34 76.66 58.60 61.38

Case I y¯1j 110.33 71.96 215.78 183.02 80.86 98.85 75.40 59.30 65.70 156.16 70.70 136.45 122.29 59.71 59.32 91.76 116.36 102.39 62.58 67.94 77.70 122.77 129.18 94.64 66.31

y¯2j 2.44 1.73 9.94 9.97 2.26 4.16 0.36 1.30 1.49 5.99 2.80 4.33 3.30 1.36 1.32 1.10 2.93 2.85 1.11 0.79 4.90 3.72 4.53 1.78 7.09

Total 1616.28 2497.44 83.55 Change (%) -15.00 -3.43 2.24

y¯2j 6.83 7.31 10.39 10.42 7.20 6.97 2.02 1.30 1.49 6.26 2.80 6.51 6.68 5.07 4.40 4.86 6.76 5.64 3.31 5.97 6.75 6.63 4.73 3.24 7.41

x¯1j 94.92 67.40 126.70 153.90 78.84 87.26 60.24 53.76 57.40 89.70 56.90 121.33 128.28 63.32 58.56 71.04 89.40 77.42 56.40 65.52 67.68 91.69 80.10 67.58 62.30

Case III y¯1j 127.41 75.20 225.50 185.60 107.48 103.30 78.80 67.35 70.33 163.20 70.70 143.32 170.08 68.02 65.12 95.90 133.52 107.00 65.40 74.64 81.20 128.30 135.00 98.90 66.70

y¯2j 8.69 7.44 10.39 10.42 8.18 7.93 3.45 3.18 3.23 6.26 2.80 9.35 9.78 7.44 6.10 7.18 8.66 6.41 5.20 7.56 7.04 6.68 4.73 3.99 7.41

1806.43 2595.42 140.94 -5.00 0.36 72.46

2027.64 6.63

2707.98 4.71

169.51 107.42

x¯1j 76.10 64.71 126.70 153.90 67.35 72.69 55.22 44.80 48.10 89.70 56.90 83.85 79.65 55.59 53.68 65.12 77.89 70.71 51.70 60.06 65.28 79.69 80.10 64.63 62.30

Case II y¯1j 115.30 75.20 225.50 185.60 84.50 103.30 78.80 59.30 65.70 163.20 70.70 142.60 127.80 63.86 63.05 95.90 121.60 107.00 65.40 71.00 81.20 128.30 135.00 98.90 66.70

Case I: Decreasing the total amount of the first input by 15% and U1 = 1.05 Case II: Decreasing the total amount of the first input by 5% and U1 = 1.1 Case III: Increasing the total amount of the first input by 10% and U1 = 1.2 Table 2 reports the results of input and output targets obtained by solving model (16). The columns of this table include three parts to show the result of case I, case II and case III, respectively. By analyzing the results in Table 2, one can find out the following conclusions. First, in case I, by solving model (14), we have δ ∗ = 0.96. It means that by decreasing total Man-hours by 15%, we can not present a target such that all output components improve, so we should 19

decrease the amount of some output at most 4% with respect to the current output level. In fact, it is reasonable that the reduction of some input resources may prevent increasing some outputs. As can be seen in the results of case I, Man-hours of the total system reduces by 15%, while the sum of Sale decreases by −3.43% and the sum of Profit increases by 2.24%. Second, in case II we have δ ∗ = 1, i.e., by decreasing the Man-hours of the total system by 5%, we can present a target such that all output components improve. In this case, the sum of Sale and the sum of Profit are increased by 0.36% and 72.46%, respectively. Contrary to the case I, both output components increased by decreasing Man-hours of the total system. Third, in case III, the total Man-hours increased by 10% while the results show that we can save (10 − 6.63)% = 3.37% of labor force such that the proposed output target is improved, as well. Forth, the results show that in cases II and III, the amount of changes in Profit is more than changing the Sale, However, in this example we assumed that w1 = w2 = 1, but one can put suitable weights for the outputs and control them. Fifth, in three cases all DMUs targets are strongly efficient. The second sample is based on 20 fast-food restaurants that are placed in the city of Hefei, Anhui Province, China, published by Du et al. (2010). These fast-food restaurants belong to the same chain store, which has a central DM to manage the decision parameters of all branches and allocate resources among them. The data set and the information of inputs and outputs are given in Table 3. Man-hours input refers to the labor used within a certain period, and shop size is the total rental floor space of the restaurant for serving the customers. The outputs are the sales of meat dishes, vegetable dishes, soups, noodles, and beverages. Note that, the only non-discretionary variable is the shop size.

20

Table 3: Inputs and Outputs data for 20 fast-food restaurants DMUs

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Total

Inputs Man-hours Shop size 103 h 102 m2 3.2 2 3.4 2.1 3.1 1.8 3.8 2.2 4.2 2.6 4.1 2.5 3.8 2.3 3.8 2.2 2.9 1.6 4.2 2.8 3.4 2.1 4 2.4 3.8 2.6 3.4 1.9 2.8 1.6 3.5 2.2 4.2 2.5 3.3 1.8 3.6 1.9 3.1 1.7 71.6 42.8

Outputs Meat dish 103 servings 2.24 2.12 2.08 2.45 2.8 2.65 2.6 2.5 2.1 2.9 2.6 2.78 2.84 2.33 2 2.4 2.68 2.05 2 2.05 48.17

Vegetable dish 103 servings 2.46 2.52 2.25 2.1 2.78 2.95 2.24 2.15 2.04 2.85 2.45 2.66 2.38 2.2 2.18 2.25 2.5 2.2 2.16 2.12 47.44

Soup 103 servings 1.22 1.34 1.05 1.3 1.42 1.38 1.15 1.1 0.98 1.52 1.36 1.18 1.25 1.06 1.96 1.26 1.46 1.12 1.02 0.94 25.07

Noodle 103 servings 3.12 3.08 2.85 2.96 3.48 3.25 3.18 3.2 2.88 3.36 3.32 3.15 3.29 2.99 2.84 2.93 3.22 3.02 2.89 2.9 61.91

Beverage 103 servings 0.96 0.88 0.74 0.79 1.05 0.98 0.95 0.82 0.72 1.12 0.82 0.98 0.85 0.82 0.71 0.74 0.92 0.78 0.74 0.68 17.05

Suppose that input and output price information is available. Thus, the outputs price vector is p = (32, 15, 10, 6, 3)T , and we assume that the inputs price vector is c = (1, 1)T . Now, we can utilize model (18) to reallocate resources. The current revenue and cost values of 20 fast-food restaurants are 2926.35 and 71.6, respectively. The central DM aims to reallocate the Man-hours among these 20 fast-food restaurants to optimize the total revenue and cost functions. For this data, we ignore the managerial constants (10) and (11). By solving model (18), the results of resource allocation are reported in Table 4.

21

DMUs

Table 4: Results of the resource reallocation for 20 fast-food restaurants. Inputs Man-hours Shop size 103 h 102 m2 3.28 2 3.4 2.1 3.04 1.8 3.56 2.2 4.2 2.6 4.04 2.5 3.72 2.3 3.56 2.2 2.8 1.6 4.2 2.8 3.4 2.1 3.88 2.4 4.2 2.6 3.16 1.9 2.8 1.6 3.56 2.2 4.04 2.5 3.04 1.8 3.16 1.9 2.92 1.7

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Total Change (%)

69.96 -2.29

42.8 0

Outputs Meat dish 103 servings 2.48 2.6 2.24 2.64 2.8 2.76 2.68 2.64 2 2.9 2.6 2.72 2.8 2.36 2 2.64 2.76 2.24 2.36 2.12

Vegetable dish 103 servings 2.4 2.45 2.29 2.52 2.78 2.71 2.58 2.52 2.18 2.85 2.45 2.65 2.78 2.34 2.18 2.52 2.71 2.29 2.34 2.23

Soup 103 servings 1.48 1.36 1.72 1.37 1.42 1.41 1.38 1.37 1.96 1.52 1.36 1.4 1.42 1.6 1.96 1.37 1.41 1.72 1.6 1.84

Noodle 103 servings 3.22 3.32 3.03 3.35 3.48 3.45 3.38 3.35 2.84 3.36 3.32 3.42 3.48 3.13 2.84 3.35 3.45 3.03 3.13 2.94

Beverage 103 servings 0.8 0.82 0.75 0.87 1.05 1 0.91 0.87 0.71 1.12 0.82 0.96 1.05 0.78 0.71 0.87 1 0.75 0.78 0.73

50.34 4.50

49.77 4.91

30.67 22.34

64.87 4.78

17.35 1.76

The results show that the total revenue is 3105.36 and increases by 3105.36 − 2926.35 = 179.01(6.12%) compared to the current total revenue. Moreover, the total cost is 69.96 and decreases by 71.6 − 69.96 = 1.64(2.3%) compared to the current total cost. Note that, in this illustrated example, we have only one re-allocatable input; however, in the application with more inputs this difference between original cost and the obtained cost will be more evident. To compare the proposed method with some resource allocation methods, we report the amount of the total productions, total revenue and cost in Table 5. Table 5: A comparison among the results of the total input and output Methods

Meat dish 103 servings Proposed 50.34 Du et al. (2010) 48.67 Lozano et al. (2011) 48.96 Fang and Li (2015) 50.34 Fang (2016) 49.47

Vegetable dish 103 servings 49.77 50.08 49.6 49.77 49.96

Soup 103 servings 30.67 33.56 30.57 30.67 32.24

Noodle 103 servings 64.87 63.36 62.62 64.77 63.99

Beverage 103 servings 17.35 17.91 17.71 17.35 17.69

Total Total Revenue Cost 3105.36 69.96 3078.185 70.862 3045.19 71.6 3105.36 71.6 3091.896 71.6

As can be seen in Table 5, the proposed method has the minimum cost and also the 22

maximum revenue among the other methods. The proposed method and Du et al.’s method (2010) have the same revenue index, but the cost in our proposed method is lower. Note that, in models (17) and (18) we aim to maximize the total revenue and minimize the total cost, respectively. Du et al. (2010) maximize the total output production and minimize the total input consumption simultaneously. Lozano et al. (2011) determine the maximum total system output increase given the current resource allocation. Fang and Li (2015) present a centralized approach for reallocating resources based on revenue efficiency and a cost-revenue analysis is performed between the reallocation cost and the revenue increase from the reallocation of inputs. Fang (2016) allocate resources to maximize the total output revenue produced by all DMUs under limited information. However, other compared methods do not consider the same objective function. Thus, further in-depth analysis and research is needed to compare our approaches with other approaches regarding the cost and revenue efficiency.

6

Conclusions

Lozano and Villa (2004) make a significant contribution to DEA literature by introducing the centralized resource allocation models. They propose a formulation that distributes inputs and outputs equally, irrespective of the units that had used or produced them. In some organizations, there is a situation in which the current resources should be increased or reduced in the next production period. These kinds of resource allocation problems can frequently be observed in practice. In this paper, we made an interpretation for Lozano and Villa’s model (2004) and expended it in order to include all manager’s requests. To fulfill our goal, we proposed two planning ideas. The first one was to maximize the total production and after that minimize the total consumption in the whole system while making all units strongly efficient. This presented method was useful in resource allocation program in the absence of information of input costs and output prices. The second idea was to maximize the total revenue then, in the second phase minimize the total cost of the organization. We proved that the first proposed centralized DEA model is a special case of our second model where the input and output prices are replaced with their absolute shadow prices. Using secondary data, we developed two empirical illustrations in Section 5. We can obtain several conclusions. First, in the first example, we assume that the market prices are not available. We consider three kinds of allocation. In the first and second cases, we allocated less amount of resources to units. The results show that by decreasing 15% of 23

the allocatable input, the amount of all outputs cannot improve. Nonetheless, by decreasing 5% of the allocatable input, we can improve the productions. This issue has not addressed sufficiently in the previous studies. In the third case, we allocated extra resources to units. In all cases, the targets of DMUs are strongly efficient (non-dominated) under the current PPS. Then, in the second example, we assumed that the market prices are available. By reallocating resources, the results showed that our method has the minimum cost and maximum revenue considering other methods.

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27

*Highlights (for review)

    

Two generalized models are presented for the centralized resource allocation. An existing resource allocation model is extended to (re)allocate some (current) additional or less resources among DMUs. The total revenue and cost are optimized and in the absence of market prices, it leads to strongly efficient targets. Two empirical examples are presented to illustrate both formulated models. The proposed method is compared to several existing methods.