A new project selection method using data envelopment analysis

Journal Pre-proofs A new project selection method using Data Envelopment Analysis Mehdi Toloo, Mahnaz Mirbolouki PII: DOI: Reference:

S0360-8352(19)30588-1 https://doi.org/10.1016/j.cie.2019.106119 CAIE 106119

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Computers & Industrial Engineering

Received Date: Revised Date: Accepted Date:

26 January 2019 5 September 2019 7 October 2019

Please cite this article as: Toloo, M., Mirbolouki, M., A new project selection method using Data Envelopment Analysis, Computers & Industrial Engineering (2019), doi: https://doi.org/10.1016/j.cie.2019.106119

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A new project selection method using Data Envelopment Analysis Mehdi Toloo a,b, Mahnaz Mirbolouki a* aFaculty bFaculty

of Economics, VŠB-Technical University of Ostrava, Ostrava, Czech Republic

of Business Administration, VŠE-University of Economics Prague, Czech Republic

Abstract The project selection problem plays a vital role in an organization to successfully attain its competitive advantages and corporate strategies. The problem is more exacerbated and compounded if the decision-maker takes the limitation of resources into consideration. As a matter of fact, the project selection problem deals with opting a set of best feasible proposals from a large pool of proposals with making the best use of available resources. It is assumed that each proposal employs various resources, such as personnel, capital, equipment, and facilities. Each subset of feasible proposals constitutes a single, composite project that utilizes a set of available but limited resources to produce various outputs. It is desired to select the best subset of proposals with the aim of using the available resources as much as possible. Data envelopment analysis (DEA) is commonly used as a prioritization method to evaluate each feasible composite project. This paper develops a new project selection method based on the performance of each contained proposals by solving a single linear DEA model. Finally, we provide a real dataset containing 21 information system proposal at Iran Ministry of Commerce to illustrate the potential application of our suggested method.

Corresponding Author (Dr. Mahnaz Mirbolouki). Emails (Mirbolouki) [email protected]; (Toloo) [email protected] ; URL: http://homel.vsb.cz/~tol0013/ *

1

Keywords: Project selection; data envelopment analysis; efficiency; information systems. Acknowledgements

The authors would like to thank the handling editor and anonymous reviewers for their valuable and constructive comments. This study was supported by the Czech Science Foundation (GAČR 19-13946S).

Highlights 

This paper studies the project selection problem.



We develop a new project selection method under resource limitations.



The advantage of the new method is that it accomplishes both individual evaluation and selection.



A case study of information system projects in Iran Ministry of Commerce validates the new method.

A new project selection method using Data Envelopment Analysis

Abstract The project selection problem plays a vital role in an organization to successfully attain its competitive advantages and corporate strategies. The problem is more exacerbated and compounded if the decision-maker takes the limitation of resources into consideration. As a matter of fact, the project selection problem deals with opting a set of best feasible proposals from a large pool of proposals with making the best use of available resources. It is assumed that each proposal employs various resources, such as personnel, capital, equipment, and facilities. Each subset of feasible proposals constitutes a single, composite project that utilizes a set of available but limited resources to produce various outputs. It is desired to select the best subset of proposals with the aim of using the available resources as much as possible. Data envelopment analysis (DEA) is commonly used as a prioritization method to evaluate each feasible 2

composite project. This paper develops a new project selection method based on the performance of each contained proposals by solving a single linear DEA model. Finally, we provide a real dataset containing 21 information system proposal at Iran Ministry of Commerce to illustrate the potential application of our suggested method. Keywords: Project selection; data envelopment analysis; efficiency; information systems. 1

introduction

The evaluation, prioritization, and selection of project proposals is a challenging issue especially in project-based organizations where it is required to assess a set of proposals competing for limited resources such as budget, equipment, and manpower. A feasible composite project involves a subset of proposals which can be established considering several undeniable restrictions on resources. Lorie & Savage (1955) introduced the capital budgeting’s problem with the aim of proposing an approach to select among competitive investment possibilities with maximum total payoff. Afterward, Weingartner (1966) extended an innovative linear programming model for addressing the project selection problem. Hitherto, a wide range of project selection applications have been investigated in the literature including construction projects (Albino & Gravelli (1998); Chang & Lee (2012); Kapelko (2018)) portfolio selection (Bhattacharyya, Kumar, & Kar (2011); Dia (2009); Jafarzadeh, Akbari, & Abedin (2018); Lim, Oh, & Zhu (2014); Pérez, Gómez, Caballero, & Liern (2018)), and Research and Development (R&D) proposals (Karasakal & Aker (2017); Khoshnevis & Teirlinck (2018); Park, Lee, & Kim (2015)). Information System (IS) is used to track, store, manipulate, and distribute required information for organizations. ISs differ from each other in terms of scope, design, and features. Effective use of ISs plays a key role in business system success. Evaluating and selecting appropriate IS projects is a challenging problem for IS managers which has drawn the attention of some scholars: Badri, Davis, & Davis (2001) formulated a goal 3

programming model with a ranking method to select a set of IS proposals that are consistent with the goals of some health service organizations. Deng & Wibowo (2008) presented a decision support system along with a multi-criteria analysis method for evaluation and selection of IS projects. Chen & Cheng (2009) developed a fuzzy multicriteria decision-making method for selecting a suitable IS project in a high-technology manufacturing company. Shakhsi-Niaei, Torabi, & Iranmanesh (2011) introduced a twophase framework based on Monte Carlo simulation for ranking and selecting IS projects under uncertainty and considering segmentation, logical, and budget constraints. Yang, Chiang, Huang, & Lin (2013) proposed a hybrid measurement model for evaluating the overall effects of IS projects based on critical, quantitative, and qualitative indicators which reflect actual business operations. Data envelopment analysis (DEA) is a non-parametric productivity analysis method that evaluates the performance of a set of homogenous decision-making units (DMUs) with multiple inputs and multiple outputs. Charnes, Cooper, & Rhodes (1978) formulated the first DEA model with the constant returns to scale (CRS) technology. The main idea behind the DEA approach is assigning a weight for each input and each output and then measuring the efficiency score of each DMU as the ratio of the weighted sum of outputs to the weighted sum of inputs. Over the last two decades, the DEA technique has gained popularity in project management as a useful mathematical tool for relative efficiency and productivity analysis. The use of DEA approach in project management goes back to Cook & Green (2000) (hereafter called CG) who investigated a DEA-based method to select a feasible composite project among a set of proposals (DMUs). Several DEAinspired approaches have been developed for addressing the project selection problem: Sowlati, Paradi, & Suld (2005) introduced an intuitive approach for prioritizing 41 IS projects of a large financial institution. Vitner, Rozenes, & Spraggett (2006) utilized the earned value management system and the multi-dimensional control systems to evaluate the performances of 11 IS projects with four elements ‘hardware’, ‘software’, ‘integration’, and ‘testing’. Kumar, Saranga, Ramírez-Márquez, & Nowicki (2007) investigated six sigma projects to identify the projects with maximum benefits. Asosheh, 4

Nalchigar, & Jamporazmey (2010) proposed an approach using a combination of balanced scorecard and DEA for evaluating three alternative information technology proposals suggested to Iran ministry of science, research, and technology. Chang & Lee (2012) modified the CG method using fuzzy numbers and developed an integrated fuzzy DEA and knapsack models in order to address the project selection problem in the engineering-procurement-construction (EPC) sector. Tavana, Khalili-Damghani, & SadiNezhad (2013) proposed a multi-objective fuzzy linear programming method to select technical projects at the National Aeronautics and Space Administration (NASA). Tavana, Keramatpour, Santos-Arteaga, & Ghorbaniane (2015) suggested a three-stage (initial screening, ranking and selecting) method to select the most suitable project portfolios in a fuzzy environment. Toloo, Nalchigar, & Sohrabi (2018) proposed a method for selecting the most efficient IS projects while taking subjective opinions and intuitive senses of decision-makers into account. More recently, Martinovic & Savic (2019) expanded the CG approach and proposed a mixed-integer linear programming model to make the most efficient assignment plan in multi-criteria, multi-project, and multi-task environments. Lozano (2013) and Wu, Zhu, Cook, & Zhu (2016) dealt with the problem of selecting the best cooperative partners. Note that only a couple of them considered resources limitations in their developed approaches. In this paper, we develop a DEA project selection model taking into account the same defined resource constraints in the CG approach with the aim of finding a feasible composite project with the highest mean-efficiency score. In contrast to the CG approach which selects a composite project with the highest aggregate-efficiency score, our suggested approach identifies a composite project with the highest mean-efficiency. The CG approach assesses virtual (aggregate) projects which their inputs and outputs are the sum of inputs and outputs of the involved proposals, respectively. However, our new approach evaluates the performance of all the individual proposals rather than evaluating virtual projects. In other words, unlike the CG approach, the performance of each individual proposal plays an important role in our new suggested approach. 5

The outline of the paper is organized as follows. Section 2 reviews the CCR model and the project selection method of CG. Section 3 develops a new project selection model. A comparative study using a real case relating to project selection in the Iranian Ministry of Commerce (IMC) is presented in Section 4 to validate the suggested method. Finally, conclusions are presented in Section 5. 2. Preliminaries In this section, we first consider the conventional radial input-oriented DEA model and then review the CG’s project selection method. 1.1 The CCR model Suppose there are 𝑛 DMUs, DMU𝑗 ( 𝑗 = 1,…,𝑛), whose performances are to be evaluated in terms of 𝑚 semi-positive inputs, 𝒙𝑗 = (𝑥1𝑗,…,𝑥𝑚𝑗), and 𝑠 semi-positive outputs, 𝒚𝑗 =

(𝑦1𝑗,…,𝑦𝑠𝑗). The relative efficiency of a unit is defined as the ratio of the weighted sum of outputs to the weighted sum of inputs. The DEA model aims at finding optimal weights for each input and output that maximizes the ratio. The following CCR model measures the efficiency score of unit under evaluation, DMU𝑜(𝑜 ∈ {1,…,𝑛}), with CRS technology: 𝑠

𝜃𝑜∗ = max ∑𝑟 = 1𝑢𝑟𝑦𝑟𝑜 s.t. 𝑚 ∑𝑖 = 1𝑣𝑖𝑥𝑖𝑜 = 1 𝑠

𝑚

∑𝑟 = 1𝑢𝑟𝑦𝑟𝑗 ― ∑𝑖 = 1𝑣𝑖𝑥𝑖𝑗 ≤ 0 𝑗 = 1,…,𝑛 𝑣𝑖 ≥ 0 𝑖 = 1,…,𝑚 𝑢𝑟 ≥ 0 𝑟 = 1,…,𝑠

(1)

where decision variables 𝑣𝑖 and 𝑢𝑟 are the weights (also referred to as multipliers) assigned to the 𝑖𝑡ℎ input and 𝑟𝑡ℎ output, respectively. The first constraint is called normalization constraint which puts an uper bound on the optimal objective value and the second constraints set ensures that the mesured efficiency will be realtive. DMU𝑜 is CCR-efficient if 𝜃𝑜∗ = 1; otherwise it is CCR-inefficient. Model (1) obtains the optimal 6

weights (𝒖 ∗ ,𝒗 ∗ ) for DMU𝑜 and hence it should be solved 𝑛 times, once per individual DMU, to measure the relative CCR-efficiency of all DMUs. 1.2 The project selection method under resource limitations In order to develop a DEA approach for selecting the best composite project, CG assumed that there are 𝑛 independent and individual proposals (DMUs) 𝑃 = {𝑝1,…,𝑝𝑛} and defined a composite project as a subset of 𝑃. As a result, any nonempty subset of 𝑃 can be considered as a single, composite project and there are 2𝑛 ―1 such composite projects. Moreover, inputs and outputs of a composite project are defined as the combined-byaddition inputs and outputs of the constituent proposals, respectively. Mathematically, let ∏(𝑃) be the power set of 𝑃 (excluding the empty set) and the subset 𝑆 = {𝑝𝑡1,…,𝑝𝑡𝑞} ∈ ∏(𝑃) be a composite project containing 𝑞 proposlas. We denote the inputs and outputs

(

𝑞

𝑞

)

(

𝑞

𝑞

)

of the composite project 𝑆 by 𝒙𝑠 = ∑𝑙 = 1𝑥1𝑡𝑙,…,∑𝑙 = 1𝑥𝑚𝑡𝑙 and 𝒚𝑠 = ∑𝑙 = 1𝑦1𝑡𝑙,…,∑𝑙 = 1𝑦𝑠𝑡𝑙 , respectively, as shown in Figure 1.

𝑥1𝑡1

𝑦1𝑡1

𝑥1𝑡1 + … + 𝑥1𝑡𝑞

𝑝𝑡1

𝑦1𝑡1 + … + 𝑦1𝑡𝑞

𝑥𝑚𝑡1

𝑦𝑠𝑡1

𝑥1𝑡𝑞

𝑦1𝑡𝑞 𝑝𝑡𝑞

𝑥𝑚𝑡1 + … + 𝑥𝑚𝑡𝑞 𝑥𝑚𝑡𝑞

𝑦𝑠𝑡𝑞

𝑦𝑠𝑡1 + … + 𝑦𝑠𝑡𝑞

Figure 1. A composite project

If there exist unlimited (enough) available resources, then a composite project containing all the proposals as the best selection for sure. However, practically, the 7

limitation of resources is a deniable fact which makes this selection more complicated. CG considered the following two main constraints for each composite project 𝑆: Resource availability (C1): The usage of each input 𝑖 of 𝑆 is less than or equal to the available resource 𝐿𝑖; i.e. ∑𝑗 ∈ 𝑆𝑥𝑖𝑗 ≤ 𝐿𝑖, 𝑖 = 1,…,𝑚. Maximum usage (C2): No project can be added to 𝑆 without violating at least one of the resource availability conditions in C1; i.e. ∀𝑝 ∈ 𝑃 ― 𝑆 ∃𝑖 s.t. ∑𝑗 ∈ 𝑆 ∪ {𝑝} 𝑥𝑖𝑗 > 𝐿𝑖 . In other words, C1 states that the sum of used resources of proposals in a composite project cannot exceed the volume of available resources. In addition, C2 express that a composite project excludes a project if and only if the amount of resource leftovers is not sufficient to support the project. Definition 1. The set 𝑆 ∈ ∏(𝑃) is called a feasible composite project if and only if it satisfies both constraints C1 and C2. Definition 2. The aggregate-efficiency of feasible composite project 𝑆 = {𝑝𝑡1,…,𝑝𝑡𝑞} is

(

𝑞

)

𝑞

defined as the CCR-efficiency of virtual DMU = (𝒙𝑠, 𝒚𝑠) where 𝒙𝑠 = ∑𝑙 = 1𝑥1𝑡𝑙,…,∑𝑙 = 1𝑥𝑚𝑡𝑙

(

𝑞

𝑞

)

and 𝒚𝑠 = ∑𝑙 = 1𝑦1𝑡𝑙,…,∑𝑙 = 1𝑦𝑠𝑡𝑙 . Assume that the feasible composite project 𝑆 is given. The following model measures the aggregate-efficiency of 𝑆: 𝑠

max ∑𝑟 = 1𝑢𝑟(∑𝑗 ∈ 𝑆𝑦𝑟𝑗) s.t. 𝑚 ∑𝑖 = 1𝑣𝑖(∑𝑗 ∈ 𝑆𝑥𝑖𝑗) = 1 𝑠 ∑𝑟 = 1𝑢𝑟𝑦𝑟𝑗

𝑣𝑖, 𝑢𝑟 ≥ 0

―

𝑚 ∑𝑖 = 1𝑣𝑖𝑥𝑖𝑗

(2) ≤ 0 𝑗 = 1,…,𝑛 𝑖 = 1,…,𝑚, 𝑟 = 1,…,𝑠

where the composite project 𝑆 with an input vector 𝒙𝑠 = ∑𝑗 ∈ 𝑆𝑥𝑖𝑗 and an output vector 𝒚𝑠 = ∑𝑗 ∈ 𝑆𝑦𝑟𝑗 plays the role of a virtual DMU. In other words, according to the DEA 8

terminology, model (2) is, in fact, the CCR model that measures the efficiency of composite project (𝒙𝑠,𝒚𝑠). As a result, in order to select the best composite project, one should solve the model (2) at most 2𝑛 ―1 times, each for a feasible composite project, which is large even for relatively modest number of proposals. For instance, if there are 20 proposals, then model (2) must be solved 524,287( = 219 ―1) times, which needs a considerable number of computations. The above approach involves two steps. The first step selects all feasible composite projects and the second step evaluates the selected composite projects. The best composite project is the one with maximum aggregate-efficiency score. However, CG formulated the following innovative mixed-binary non-linear programming (MBNLP) model which combines both selection and evaluation steps into a single step: 𝑠

(

)

𝑛

𝑧 ∗ = max ∑𝑟 = 1𝑢𝑟 ∑𝑗 = 1𝑐𝑗𝑦𝑟𝑗 s.t. 𝑚 𝑛 ∑𝑖 = 1𝑣𝑖 ∑𝑗 = 1𝑐𝑗𝑥𝑖𝑗 = 1

(

)

𝑠 𝑚 ∑𝑟 = 1𝑢𝑟𝑦𝑟𝑗 ― ∑𝑖 = 1𝑣𝑖𝑥𝑖𝑗 𝑛 ∑𝑗 = 1𝑐𝑗𝑥𝑖𝑗 + 𝑙𝑖 = 𝐿𝑖

≤0

(3.𝑎) 𝑗 = 1,…,𝑛

(3.𝑏)

𝑖 = 1,…,𝑚 (1 ― 𝑐𝑗)𝑥𝑖𝑗 + 𝑀𝑐𝑗 + 𝑀𝑑𝑖𝑗 ≥ 𝑙𝑖 + 𝜀 𝑗 = 1,…,𝑛,𝑖 = 1,…,𝑚 𝑚 ∑𝑖 = 1𝑑𝑖𝑗 ≤ 𝑚 ― 1 𝑗 = 1,…,𝑛 𝑗 = 1,…,𝑛,𝑖 = 1,…,𝑚 𝑐𝑗,𝑑𝑖𝑗 ∈ {0,1} 𝑗 = 1,…,𝑛,𝑖 = 1,…,𝑚, 𝑟 = 1,…,𝑠 𝑣𝑖, 𝑢𝑟,𝑙𝑖 ≥ 0

(3.𝑐) (3.𝑑) (3.𝑒) (3.𝑓) (3.𝑔)

(3)

where 𝜀 > 0 is a very small number and 𝑀 is a sufficiently large number. The indicator binary variable† 𝑐𝑗 is introduced for the 𝑗𝑡ℎ proposal such that if 𝑐𝑗∗ = 1, then 𝑝𝑗 is included into the composite project; otherwise, it is excluded from the compsoite project. Also, 𝑙𝑖 is the slack in 𝑖𝑡ℎ available resource; 𝑙𝑖 = 0 indicates that there is no available 𝑖𝑡ℎ

†

An indicator variable is a binary variable which indicates a certain state in a model.

9

𝑛

resource meanwhile 𝑙𝑖 > 0 shows that 𝐿𝑖 ― ∑𝑗 = 1𝑐𝑗𝑥𝑖𝑗 amount of the 𝑖𝑡ℎ resource is available. 𝑑𝑖𝑗 is a binary variable appears in the constraints (3.𝑑) and (3.𝑒) to meet the condition C2. The optimal objective function value 𝑧 ∗ of model (3) shows the highest aggregate-efficiency score of selected feasible composite project. In order to analyze the model, we consider the following two cases: Case A (𝑐𝑗 = 1). In this case (i) the contribution of the 𝑗𝑡ℎ proposal in the objective 𝑠

function is ∑𝑟 = 1𝑢𝑟𝑦𝑟𝑗 which indicates its contribution in efficiency score of the composite 𝑚

project; (ii) the contribution of the 𝑗𝑡ℎ proposal in the normalization constraint is ∑𝑖 = 1𝑣𝑖 𝑥𝑖𝑗; (iii) constraints set (3.𝑐) assigns (𝑥1𝑗,….,𝑥𝑚𝑗) resources to the 𝑗𝑡ℎ proposal; (iv) the constraint (1 ― 𝑐𝑗)𝑥𝑖𝑗 +𝑀𝑐𝑗 +𝑀𝑑𝑖𝑗 ≥ 𝑙𝑖 +𝜀 is a redundant constraint 𝑀(1 + 𝑑𝑖𝑗) ≥ 𝑙𝑖 +𝜀 𝑙𝑖 + 𝜀

because 𝑀 is a sufficiently large number, e.g. 𝑀 ≥ 1 + 𝑑𝑖𝑗; (v) 𝑑𝑖𝑗 binary variables are free to get either zero or one value, however, according to constraint (3.𝑒), all the binary variables 𝑑1𝑗,…,𝑑𝑚𝑗 cannot get positive values simultaneously. Note that the redundancy of constraint (3.𝑒) follows form the redundant constraints set (3.𝑑). Case B (𝑐𝑗 = 0). (i) the contributions of 𝑗𝑡ℎ proposal in the objective function, i.e.,

𝑐𝑗

(∑𝑟𝑠= 1𝑢𝑟𝑦𝑟𝑗), and in the normalization constraint, i.e., 𝑐𝑗(∑𝑖𝑚= 1𝑣𝑖𝑥𝑖𝑗), are zero; (ii) no resource will be assigned to the 𝑗𝑡ℎ proposal due to the constraints set (3.𝑐); (iii) constraints set (3.𝑒) guarantees that at least one of the variables 𝑑1𝑗,𝑑2𝑗,…,𝑑𝑚𝑗 remains zero. Without loos of genearlaity we let 𝑑𝑘𝑗 = 0; (iv) the constraints set (3.𝑑) for 𝑖 = 𝑘 turns to 𝑥𝑘𝑗 ≥ 𝑙𝑘 +𝜀 or equvalenlty 𝑥𝑘𝑗 > 𝑙𝑘, which means that the remaining amount of 𝑘𝑡ℎ resource, i.e. 𝑙𝑘, is not sufficient for assigning to the 𝑗𝑡ℎ proposal. It should be highlighted here that in model (3) the efficiency of the 𝑗𝑡ℎ proposal is defined 𝑠

as 𝐸𝑗 =

∑𝑟 = 1𝑢𝑟𝑦𝑟𝑗 𝑚

∑𝑖 = 1𝑣𝑖𝑥𝑖𝑗

and hence the constraints set (3.𝑏) keeps this ratio less than or eqaul to

one for all the proposals (selected or not) in the evaluation of the compiste project. 10

CG utilized the change of variables 𝑎𝑟𝑗 = 𝑢𝑟𝑐𝑗 and 𝑏𝑖𝑗 = 𝑣𝑖𝑐𝑗 and formulated the following equivalent mixed binary linear programming (MBLP) model: 𝑠

𝑛

𝑧 ∗ = max ∑𝑟 = 1∑𝑗 = 1𝑎𝑟𝑗𝑦𝑟𝑗 s.t. 𝑚 𝑛 ∑𝑖 = 1∑𝑗 = 1𝑏𝑖𝑗𝑥𝑖𝑗 = 1 𝑠

𝑚

∑𝑟 = 1𝑢𝑟𝑦𝑟𝑗 ― ∑𝑖 = 1𝑣𝑖𝑥𝑖𝑗 ≤ 0 𝑛 ∑𝑗 = 1𝑐𝑗𝑥𝑖𝑗

+ 𝑙𝑖 = 𝐿𝑖 (1 ― 𝑐𝑗)𝑥𝑖𝑗 + 𝑀𝑐𝑗 + 𝑀𝑑𝑖𝑗 ≥ 𝑙𝑖 + 𝜀 𝑚 ∑𝑖 = 1𝑑𝑖𝑗 ≤ 𝑚 ― 1 0 ≤ 𝑎𝑟𝑗 ≤ 𝑀𝑐𝑗 0 ≤ 𝑏𝑖𝑗 ≤ 𝑀𝑐𝑗 𝑢𝑟 ≥ 𝑎𝑟𝑗 𝑏𝑖𝑗 ≤ 𝑣𝑖 ≤ 𝑏𝑖𝑗 + 𝑀(1 ― 𝑐𝑗) 𝑐𝑗,𝑑𝑖𝑗 ∈ {0,1} 2

𝑗 = 1,…,𝑛 𝑖 = 1,…,𝑚 𝑗 = 1,…,𝑛,𝑖 = 1,…,𝑚 𝑗 = 1,…,𝑛 𝑗 = 1,…,𝑛,𝑟 = 1,…,𝑠 𝑗 = 1,…,𝑛,𝑖 = 1,…,𝑚 𝑗 = 1,…,𝑛,𝑟 = 1,…,𝑠 𝑗 = 1,…,𝑛,𝑖 = 1,…,𝑚 𝑗 = 1,…,𝑛,𝑖 = 1,…,𝑚

(4)

The proposed new project selection method

The CG approach considers the composite project as a virtual DMU and hence the 𝑛

aggregate-efficiency is measured by maximizing the weighted sum of ∑𝑗 = 1𝑦𝑟𝑗 (𝑟 = 1,…,𝑠) 𝑛

over the weighted sum of ∑𝑗 = 1𝑥𝑖𝑗 (𝑖 = 1,…,𝑚). In other words, the composite project is treated as the (𝑛 + 1)𝑡ℎ proposal which is under assessment unit and the CG approach, as a common-weights approach, finds the optimal common set of weights (𝒗 ∗ ,𝒖 ∗ ). As a result, in the CG approach the CCR-efficiency score of selected proposals do not play a key role in the assessment. Reference to Cook & Zhu (2007) reveals that the CG approach (as an aggregate approach) might be very sensitive to extreme proposals: ‘… it is recognized that the aggregate approach might be criticized for being overly sensitive to extreme DMUs, or possibly to the ‘larger’ DMUs. Such arguments are common in the MCDM literature, where a consensus ranking among responses from a collection of voters is to be derived. See, for example, Cook (2006). ‘ 11

We develop an alternative approach to deal with the project selection problem based on the individual CCR-efficiency score of the selected proposals. Toward this end, we define the efficiency of a composite project as the mean of selected proposals. Definition 3. The mean-efficiency of feasible composite project 𝑆 = {𝑝𝑡1,…,𝑝𝑡𝑞} is the arithmetic mean of CCR-efficiency score of the proposals 𝑝𝑡1,…,𝑝𝑡𝑞. Figure 2 makes a comparison between the CG and our approaches. The CG approach first assigns a single common input-weight 𝑣𝑖 and output-weight 𝑢𝑟 to the 𝑖𝑡ℎ aggregate input 𝑞

𝑞

∑𝑙 = 1𝑥𝑖𝑡𝑙 for 𝑖 = 1,...,𝑚 and 𝑟𝑡ℎ aggregate output ∑𝑙 = 1𝑦𝑟𝑡𝑙 for 𝑟 = 1,…,𝑠, respectively, and

( then maximizes 𝑣1(∑

𝑞

) ( 𝑥1𝑡𝑙) + … + 𝑣𝑚(∑

) . However, our new approach firstly devotes an )

𝑞

𝑢1 ∑𝑙 = 1𝑦1𝑡𝑙 + … + 𝑢𝑠 ∑𝑙 = 1𝑦𝑠𝑡𝑙 𝑞 𝑙=1

𝑞 𝑥 𝑙 = 1 𝑚𝑡𝑙

input-weight vector (𝑣𝑖𝑡1,…,𝑣𝑖𝑡𝑞) and an output-weight vector (𝑢𝑟𝑡1,…,𝑢𝑟𝑡𝑞) to the 𝑖𝑡ℎ composite input (𝑥𝑖𝑡1,…,𝑥𝑖𝑡𝑞) and 𝑟𝑡ℎ composite output (𝑦𝑟𝑡1,…,𝑦𝑟𝑡𝑞), respectively, and then 𝑞

it maximizes

𝑞

∑𝑙 = 1(𝑢1𝑡 𝑦1𝑡 ) + … + ∑𝑙 = 1(𝑢𝑠𝑡 𝑦𝑠𝑡 ) 𝑙

𝑙

𝑙

𝑞

𝑙

𝑞

∑𝑙 = 1(𝑣1𝑡 𝑥1𝑡 ) + … + ∑𝑙 = 1(𝑣𝑚𝑡 𝑥𝑚𝑡 ) 𝑙

𝑙

𝑙

𝑠

which is equivalent to

𝑙

𝑚

we let ∑𝑖 = 1𝑣𝑖𝑡𝑙𝑥𝑖𝑡𝑙 = 1 for 𝑙 = 1,…,𝑞.

12

𝑠

∑𝑟 = 1𝑢𝑟𝑡 𝑦𝑟𝑡 + … + ∑𝑟 = 1𝑢𝑟𝑡 𝑦𝑟𝑡 1 𝑞 1 𝑞 𝑞

if

𝑥1𝑡1

(

𝑞

)

𝑣1 ∑𝑙 = 1𝑥1𝑡𝑙

𝑦1𝑡1 𝑝𝑡1 𝑦𝑠𝑡1

𝑥𝑚𝑡1

𝑥1𝑡𝑞

(

𝑞

)

𝑣𝑚 ∑𝑙 = 1𝑥𝑚𝑡𝑙

(

𝑞

)

(

𝑞

)

𝑢1 ∑𝑙 = 1𝑦1𝑡𝑙

𝑦1𝑡𝑞 𝑝𝑡𝑞

𝑥𝑚𝑡𝑞

𝑦𝑠𝑡𝑞

𝑢𝑠 ∑𝑙 = 1𝑦𝑠𝑡𝑙

a. CG approach (Common-weights)

𝑥1𝑡1 𝑝𝑡1

𝑞

∑𝑙 = 1(𝑣1𝑡𝑙𝑥1𝑡𝑙)

𝑞

𝑥𝑚𝑡1

𝑦𝑠𝑡1

𝑥1𝑡𝑞

𝑦1𝑡𝑞 𝑝𝑡𝑞

𝑞

∑𝑙 = 1(𝑣𝑚𝑡𝑙𝑥𝑚𝑡𝑙)

𝑦1𝑡1

𝑥𝑚𝑡𝑞

∑𝑙 = 1(𝑢1𝑡𝑙𝑦1𝑡𝑙)

𝑞

𝑦𝑠𝑡𝑞

∑𝑙 = 1(𝑢𝑠𝑡𝑙𝑦𝑠𝑡𝑙)

b. New approach Figure 2. aggregate-efficiency vs. mean-efficiency

13

In this section, we develop a new project selection model which analogous to the CG approach accomplishes both individual evaluation and selection by solving a single model, nevertheless, in contrast to the CG approach, it assigns an individual weight for each input and outputs of each selected proposals. Before proceeding we utilize the following model of Beasley (2003) which consolidates 𝑛 independent linear programs with the aim of maximizing the mean-efficiency score of all obervations simultaneously through a single model: 𝑛

𝑠

∑𝑗 = 1∑𝑟 = 1𝑢𝑟𝑗𝑦𝑟𝑗

𝛩 = max 𝑛 s.t. 𝑚 ∑𝑖 = 1𝑣𝑖𝑗𝑥𝑖𝑗 = 1 𝑠 ∑𝑟 = 1𝑢𝑟𝑗𝑦𝑟𝑡

𝑣𝑖𝑗 ≥ 0 𝑢𝑟𝑗 ≥ 0

―

𝑗 = 1,…,𝑛

𝑚 ∑𝑖 = 1𝑣𝑖𝑗𝑥𝑖𝑡

(5)

≤ 0 𝑗, 𝑡 = 1,…,𝑛 𝑗 = 1,…,𝑛, 𝑖 = 1,…,𝑚 𝑗 = 1,…,𝑛, 𝑟 = 1,…,𝑠

where 𝑣𝑖𝑗 and 𝑢𝑟𝑗 represent the 𝑖𝑡ℎ input and 𝑟𝑡ℎ output weight for DMU𝑗 (𝑗 = 1,…,𝑛). In order to compare and contrast models (1) and (5), let 𝐹𝑜 and 𝐹 be the feasible region of these models, respectively, i.e., 𝐹𝑜 = {(𝒗,𝒖)│𝒗𝒙𝑜 = 1, 𝒖𝐘 ― 𝒗𝐗 ≤ 𝟎𝑛, 𝒗 ≥ 𝟎𝑚, 𝒖 ≥ 𝟎𝑠} ⊂ ℝ𝑚 + 𝑠 𝐹 = {(𝐕 ,𝐔)│𝐕𝐗 = 𝟏𝑛, 𝐔𝐘 ― 𝐕𝐗 ≤ 𝟎𝑛2,𝐕 ≥ 𝟎𝑚 × 𝑛, 𝐔 ≥ 𝟎𝑠 × 𝑛 } ⊂ ℝ𝑛 × (𝑚 + 𝑠) where 𝑿 and 𝒀 are the input and output matrixes , respectively, i.e.,

[

𝑥11 𝑥12 𝑥 𝑥22 𝐗 = ⋮21 ⋮ 𝑥𝑚1 𝑥𝑚2

] [

… 𝑥1𝑛 𝑦11 𝑦12 … 𝑥2𝑛 𝑦21 𝑦22 ⋱ ⋮ ,𝒀= ⋮ ⋮ … 𝑥𝑚𝑛 𝑦𝑠1 𝑦𝑠2

14

]

… 𝑦1𝑛 … 𝑦2𝑛 ⋱ ⋮ , … 𝑦𝑠𝑛

𝒗 = (𝑣1,…,𝑣𝑚) and 𝒖 = (𝑢1,…,𝑢𝑠) are the input and output weights, 𝟎𝑛 represents the zero vector‡, 𝟏𝑛 stands for the sum vector§, 𝐕 and 𝐔 are the input and output weights matrixes, respectivly, i.e.,

[

𝑣11 𝑣12 𝑣21 𝑣22 𝐕= ⋮ ⋮ 𝑣𝑚1 𝑣𝑚2

] [

𝑢11 𝑢12 … 𝑣1𝑛 … 𝑣2𝑛 𝑢21 𝑢22 ⋱ ⋮ , 𝐔= ⋮ ⋮ … 𝑣𝑚𝑛 𝑢𝑠1 𝑢𝑠2

]

… 𝑢1𝑛 … 𝑢2𝑛 ⋱ ⋮ , … 𝑢𝑠𝑛

and 𝟎𝑚 × 𝑛 denotes the zero matrix in ℝ𝑚 × 𝑛 space.

[]

𝐹1 It is easy to verify that 𝐹 = ⋮ . Let (𝐕 ∗ ,𝐔 ∗ ) ∈ ℝ𝑛+× (𝑚 + 𝑠) be the optimal solution for 𝐹𝑛 model (5). As inspectaion makes clear, the 𝑜𝑡ℎ row of the optimal matrix (𝐕 ∗ ,𝐔 ∗ ) is the optimal solution of the CCR model (1) when DMU𝑜 is under evalaution and hence 𝛩 = 𝑛

𝜃𝑗∗

∑𝑗 = 1 𝑛 . Let us proceed with our earlier assumption that there are 𝑛 independent proposals under consideration with 𝑚 inputs and 𝑟 outputs. We suggest the following extention of model (5) which selects the best feasible composite project by maximizing the meanefficiency of selected proposlas:

‡ §

A vector with all components equal to zero or the origin in ℝ𝑛 space A vector having each component equal to 1.

15

𝑛

∗

𝜑 = max

𝑠

∑𝑗 = 1∑𝑟 = 1𝑢𝑟𝑗𝑦𝑟𝑗 𝑛

s.t. 𝑚 ∑𝑖 = 1𝑣𝑖𝑗𝑥𝑖𝑗 = 𝑐𝑗

∑𝑗 = 1𝑐𝑗

𝑠 𝑚 ∑𝑟 = 1𝑢𝑟𝑗𝑦𝑟𝑡 ― ∑𝑖 = 1𝑣𝑖𝑗𝑥𝑖𝑡 𝑛 ∑𝑗 = 1𝑐𝑗𝑥𝑖𝑗 + 𝑙𝑖 = 𝐿𝑖

≤0

𝑗 = 1,…,𝑛

(6.𝑎)

𝑗, 𝑡 = 1,…,𝑛

(6.𝑏)

𝑖 = 1,…,𝑚 (1 ― 𝑐𝑗)𝑥𝑖𝑗 + 𝑀𝑐𝑗 + 𝑀𝑑𝑖𝑗 ≥ 𝑙𝑖 + 𝜀 𝑗 = 1,…,𝑛,𝑖 = 1,…,𝑚 𝑚 ∑𝑖 = 1𝑑𝑖𝑗 ≤ 𝑚 ― 1 𝑗 = 1,…,𝑛 𝑗 = 1,…,𝑛,𝑟 = 1,…,𝑠 0 ≤ 𝑢𝑟𝑗 ≤ 𝑀𝑐𝑗 0 ≤ 𝑣𝑖𝑗 ≤ 𝑀𝑐𝑗 𝑗 = 1,…,𝑛,𝑖 = 1,…,𝑚 𝑙𝑖 ≥ 0 𝑖 = 1,…,𝑚 𝑐𝑗,𝑑𝑖𝑗 ∈ {0,1} 𝑗 = 1,…,𝑛,𝑖 = 1,…,𝑚

(6)

(6.𝑐) (6.𝑑) (6.𝑒) (6.𝑓) (6.𝑔)

In this model, the auxiliary binary variable 𝑐𝑗 is introduced for the 𝑗𝑡ℎ proposal. To clarify the model formulation, some explanations are provided: The constraints set (6.𝑐) ― (6.𝑒) are adapted from model (3) to establish the feasiblity of selected proposals. In this model, if 𝑐𝑗 = 0, then from the last two constaints we get 𝑣𝑖𝑗 = 𝑢𝑟𝑗 = 0, ∀𝑖, ∀𝑟 which indicates that the 𝑗𝑡ℎ proposal is abandoned from the evaluation; otherwise, the weights 𝑣𝑖𝑗, ∀𝑖, and 𝑢𝑟𝑗 , ∀𝑟 can achieve non-negative values. Hence, the constraints sets (6.𝑎) and (6.𝑏) are 𝑠

active and the share of the 𝑗𝑡ℎ proposal in the objective fucntion is ∑𝑟 = 1𝑢𝑟𝑗𝑦𝑟𝑗. In addition, 𝑚

for each selected proposal there is a normalization constaint ∑𝑖 = 1𝑣𝑖𝑗𝑥𝑖𝑗 = 1. Suppose that our suggested MBNLP model (6) is solved and the optimal solution (𝒗1∗ ,..𝒗𝑚∗ ∗ ∗) ,𝒖1∗ ,…,𝒖𝑠∗ ,𝒄 ∗ ,𝒅1∗ ,…,𝒅𝑚∗ )where 𝒗𝑖∗ = (𝑣𝑖1∗ ,…,𝑣𝑖𝑛∗ ); 𝑖 ∈ {1,…,𝑚}, 𝒖𝑠∗ = (𝑢𝑟1 ; 𝑟 ∈ {1,…,𝑠} ,…,𝑢𝑟𝑛

, 𝒄 ∗ = (𝑐1∗ ,…,𝑐𝑛∗ ) and 𝒅𝑖∗ = (𝑑𝑖1∗ ,…,𝑑𝑖𝑛∗ );𝑖 ∈ {1,…,𝑚} are at hand. The best subset including the selected projects through model (6) is determined as 𝑆 ∗ = {𝑝𝑗│𝑐𝑗∗ = 1, 𝑗 = 1,…,𝑛}. Additionally, the CCR-efficiency score of the 𝑗𝑡ℎ selected proposal (𝑝𝑗 ∈ 𝑆 ∗ ) can be 𝑠

obtained as 𝜃𝑝∗𝑗 = ∑𝑟 = 1𝑢𝑟𝑗∗ 𝑦𝑟𝑗. The following theorem makes a comparison between models (3) and (6).

16

Theorem 1. Let (𝒗 ∗ ,𝒖 ∗ ,𝒍 ∗ ,𝒄 ∗ ,𝒅 ∗ ) be the optimal solution for model (3) and 𝐽 ∗ be the index set of selected proposals. (𝑽 ,𝑼,𝒍 ∗ ,𝒄 ∗ ,𝒅 ∗ ) is a feasible solution for model (6) where

{

𝒗∗

{

∗

𝒖∗

if 𝑗 ∈ 𝐽 if 𝑗 ∈ 𝐽 ∗ 𝒗𝑗 = 𝒗 ∗ 𝒙𝑗 & 𝒖𝑗 = 𝒗 ∗ 𝒙𝑗 𝟎𝑚 otherwise 𝟎𝑠 otherwise Proof. From the constraint (3.𝑎), 𝒗 ∗ (∑𝑗 ∈ 𝐽 ∗ 𝒙𝑗) = 1, we obtain the constraint (6.𝑎) 𝑚

∑𝑖 = 1𝑣𝑖𝑗𝑥𝑖𝑗 = 𝑐𝑗∗ , ∀𝑗. Moreover, dividing the constraints (3.𝑏) by 𝒗 ∗ 𝒙𝑗 for each 𝑗 ∈ 𝐽 ∗ results in the constraints (6.𝑏). The next three constraints are common in both models. It is easy to verify that the constraints (6.𝑓) and (6.𝑔) are hold when one selects 𝑀 ≥ max

{

𝒗𝑖∗ 𝒗

∗

𝒖𝑟∗

}

, ;∀𝑗 ∈ 𝐽 ∗ ,∀𝑖 , ∀𝑟 which completes the proof. ■ 𝒙 𝒗∗𝒙 𝑗

𝑗

Model (6) is non-linear due to division in its objective function. In order to eliminate the nonlinear term from the model formulation, we suggest the following change of variables 𝑤=

1 𝑛 ∑𝑗 = 1𝑐𝑗

,

𝑐𝑗𝑤 = 𝑘𝑗,

𝑣′𝑖𝑗 = 𝑣𝑖𝑗𝑤 and

𝑢′𝑟𝑗 = 𝑢𝑟𝑗𝑤 (𝑗 = 1,…,𝑛,𝑖 = 1,…,𝑚,𝑟 = 1,…,𝑠)

substitute 𝑣′𝑖𝑗 and 𝑢′𝑟𝑗 for 𝑣𝑖𝑗 and 𝑢𝑟𝑗, respectively, to formualte the following MBLP:

17

and

𝑛

𝑠

𝜑 ∗ = max (∑𝑗 = 1∑𝑟 = 1𝑢𝑟𝑗𝑦𝑟𝑗) s.t. 𝑠 𝑚 ∑𝑟 = 1𝑢𝑟𝑗𝑦𝑟𝑡 ― ∑𝑖 = 1𝑣𝑖𝑗𝑥𝑖𝑡 ≤ 0 𝑚 ∑𝑖 = 1𝑣𝑖𝑗𝑥𝑖𝑗 = 𝑘𝑗 𝑛 ∑𝑗 = 1𝑐𝑗𝑥𝑖𝑗 + 𝑙𝑖 =

𝐿𝑖 (1 ― 𝑐𝑗)𝑥𝑖𝑗 + 𝑀𝑐𝑗 + 𝑀𝑑𝑖𝑗 ≥ 𝑙𝑖 + 𝜀 𝑚 ∑𝑖 = 1𝑑𝑖𝑗 ≤ 𝑚 ― 1 0 ≤ 𝑢𝑟𝑗 ≤ 𝑀𝑐𝑗 0 ≤ 𝑣𝑖𝑗 ≤ 𝑀𝑐𝑗 𝑛 ∑𝑗 = 1𝑘𝑗 = 1 𝑘𝑗 ≤ 𝑐𝑗 𝑤 ― 𝑘𝑗 ≤ 𝑀(1 ― 𝑐𝑗) 𝑘𝑗 ≤ 𝑤 𝑙𝑖 ≥ 0, 𝑤 ≥ 0 𝑐𝑗,𝑑𝑖𝑗 ∈ {0,1}

𝑗, 𝑡 = 1,…,𝑛 𝑗 = 1,…,𝑛 𝑖 = 1,…,𝑚 𝑗 = 1,…,𝑛,𝑖 = 1,…,𝑚 𝑗 = 1,…,𝑛 𝑗 = 1,…,𝑛,𝑟 = 1,…,𝑠 𝑗 = 1,…,𝑛,𝑖 = 1,…,𝑚

(7)

𝑗 = 1,…,𝑛 𝑗 = 1,…,𝑛 𝑗 = 1,…,𝑛 𝑖 = 1,…,𝑚 𝑗 = 1,…,𝑛,𝑖 = 1,…,𝑚

In general, model (7) finds the best composite project that not only meets the conditions C1 and C2 but also contains the proposals with the highest mean-efficiency score. The main purpose of our approach is to take the individual CCR-efficiency scores into account and select projects in terms of better performance as much as possible. In order to compare the size of models (7) and (4) we should highlight that the former model has (𝑛(𝑚 + 𝑠 + 1) + 𝑚 + 1) non-negative continuous varivbles, 𝑛(𝑚 + 1) binary variables, and (𝑛2 + 𝑛(2𝑚 + 𝑠 + 5) + 𝑚 + 1)constraints, menwhile, the latter model posseess (𝑛 + 1)(𝑚 + 𝑠) non-negative continuous varaibles, 𝑛(𝑚 + 1) binary variables, and (𝑛(4𝑚 + 2𝑠 + 2) + 1) constraints. In other word, both models include the same number of binary variables and model (7) has (𝑛 ― 𝑠 + 1) continuous variables and

((𝑛2 + 3𝑛 + 𝑚) ― (2𝑛𝑚 + 𝑛𝑠)) constraints more than model (4). The following section illustrates the applicability and efficacy of the proposed method and makes a comparison between the proposed method and the CG method. 3. Application

18

Nowadays, organizations need an IS to survive and thrive. An IS is defined as a set of interrelated elements that conduct the tasks of collecting, processing, storing, and distributing information to support decision making and control in an organization. ISs eliminate geographical boundaries for organizations and provide them ample opportunities to develop new products, services, availability and etc. Implementation of any IS differs from others in terms of scope, design, and features. A key component of IS management is evaluating and selecting efficient IS projects which have been proposed to an organization. Iran Ministry of Commerce (IMC) aims at managing the regulation and implementation of policies applicable to domestic and foreign trade. IS proposals are periodically submitted to IMC and the IS manager used to perform the task of selecting appropriate projects. The manager had to fulfill this selection depends on experience and intuition. As a matter of fact, prioritizing and selecting projects in the absence of a systematic procedure is a challenging issue which may prevent IMC from achieving its goals and objectives. Hence, it is a need for developing a model which can be used as an optimization tool for evaluating and selecting the suitable IS proposals. IMC deals with financial problems for approving proposals which should be considered in the evaluation and selection. IS department of IMC employs a research team including a couple of IS specialists in order to develop some key criteria for evaluating the projects. Each IS proposal improves the process of directing and controlling (managing) a business as a set of capabilities which describes what a business does. ‘Time reduction’ and ‘improvement management capabilities’ are two criteria for indicating acceleration of the process and enhancement of IMC capabilities such as innovation, competitiveness, and survive, respectively. The IS department of IMC evaluates each proposal with five criteria: the costs of software (𝑥1), training (𝑥2), and supporting (𝑥3) along with time reduction (𝑦1), and improvement management capabilities (𝑦2). The smaller input and larger output amounts are preferable. It should be noted here that the inputs are directly obtained from the 19

proposals meanwhile the outputs are subjectively scored by specialists and experts in IMC. Table 1 exhibits the input and output data of 21 IS submitted proposals to IMC. The CCR-efficiency score of each proposal is reported in the column labeled “CCR” and the last two columns point out the optimal indicator binary variable 𝑐𝑗∗ measured by the CG and our approaches, respectively. In addition, the last row shows the avaiable resources along with the measured aggregate- and mean-efficiency scores for each selection. Table 1. Data and results NEW

proposals

𝑥1

𝑥2

𝑥3

𝑦1

𝑦2

CCR

CG

𝒑𝟏

4764

183

118

21

7

0.95

1

1

𝒑𝟐

3552

205

140

13

7

0.85

0

0

𝒑𝟑

5813

116

98

12

3

0.63

1

0

𝒑𝟒

3327

125

82

17

5

0.98

1

1

𝒑𝟓

3286

184

76

19

3

1

0

1

𝒑𝟔

4714

177

117

22

6

0.69

0

0

𝒑𝟕

3020

116

93

12

5

1

1

1

𝒑𝟖

6110

291

102

16

3

0.63

0

0

𝒑𝟗

4834

178

232

11

8

1

1

1

𝒑𝟏𝟎

2100

128

279

19

4

1

0

1

𝒑𝟏𝟏

3436

190

128

31

8

1

1

1

𝒑𝟏𝟐

5092

197

152

30

7

0.93

1

1

𝒑𝟏𝟑

4585

218

99

11

3

0.49

1

0

𝒑𝟏𝟒

3801

209

143

19

2

0.56

1

0

𝒑𝟏𝟓

5478

112

106

10

5

1

0

1

𝒑𝟏𝟔

7042

218

129

15

1

0.48

0

0

𝒑𝟏𝟕

6175

147

185

21

3

0.88

0

0

𝒑𝟏𝟖

4083

198

132

11

5

0.61

0

0

𝒑𝟏𝟗

5173

201

187

21

7

0.70

0

0

𝒑𝟐𝟎

4217

196

107

13

6

0.89

0

1

𝒑𝟐𝟏

3842

129

165

12

3

0.57

0

0

40000

2400

2000

Aggregate-efficiency

0.898

0.839

Mean-efficiency

0.943

0.975

Available Resource

20

Method

Reference to Table 1 shows that the CG approach selects the composite project 𝐶𝑃𝐶𝐺 = {𝑝1,𝑝3,𝑝4,𝑝7,𝑝9,𝑝11,𝑝12,𝑝13,𝑝14} while our new approach opts different composite project 𝐶𝑃𝑛𝑒𝑤 = {𝑝1,𝑝4,𝑝5,𝑝7,𝑝9,𝑝10,𝑝11,𝑝12, 𝑝15,𝑝20}. Both composite projects involve six common proposals 𝑝1, 𝑝4, 𝑝7, 𝑝9, 𝑝11and 𝑝12. There are three CCR-efficient proposals in 𝐶𝑃𝐶𝐺 however all the six CCR-efficient proposlas are included in 𝐶𝑃𝑛𝑒𝑤. The minimum CCReffciency score of proposals in the former compsoite project is 0.49 while those in the latter compiste project is 0.89. 𝐶𝑃𝐶𝐺 involves nine proposals whereas 𝐶𝑃𝑛𝑒𝑤 includes 10 proposals. The CG approach evalutes the CCR-efficiency of virtual project (𝑿𝐶𝐺,𝒀𝐶𝐺) where 𝑿𝐶𝐺 = (37864,1345,1288), 𝒀𝐶𝐺 = (163, 52)**. The mean-efficiency of 𝐶𝑃𝐶𝐺 can be easily measured as the arithmatic mean of CCR-efficiency scores of selected proposals††, which is equal to 0.943. The aggreagte-efficiency of 𝐶𝑃𝑛𝑒𝑤 equals to 0.840 which is obatined by solving model (2) where 𝑆 = 𝐶𝑃𝑛𝑒𝑤. An easy computation revelas that 𝐶𝑃𝑛𝑒𝑤 produces 𝒀𝑛𝑒𝑤 = (183,58) outputs which clearly dominates 𝒀𝐶𝐺. In other words, the new aprroach results in more time redution and more improvement managemnt capabilities in comparision with the CG apprach. In order to have a more detailed analysis, we calculate the remaining resources through each method. The new approach costs 𝑿𝑛𝑒𝑤 = (39554,1609,1373) for software, tarining, and supporting for the selected proposals. Figure 3 compares the remaining resources of each approach which points out that the new approach efficiently employes the

**

their components are the sum of inputs and outputs of

respectively. ††

i.e.,

(𝜃1∗ + 𝜃3∗ + 𝜃4∗ + 𝜃7∗ + 𝜃9∗ + 𝜃11∗ + 𝜃12∗ + 𝜃13∗ + 𝜃14∗ ) 9

.

21

𝑝1, 𝑝3, 𝑝4, 𝑝7, 𝑝9, 𝑝11, 𝑝12, 𝑝13 and 𝑝14,

avalaible and limited resourcses in order to produce more outputs. Note that the remaing software cost in our approach is 79% less than the CG approach.

2500 2136 2000

1500 1055 1000

791

712

627

446

500

0 Software cost

Training cost CG

Support cost

New model

Figure 3. Remaining resources in each method

Moreover, Table 2 exhibits the optimal common set of weights obtained by the CG approach along with the optimal weights relates to each selected individual proposal resulted by our approach. The unselected proposals by our approach possess zero inputoutput weights, i.e., (𝑣1𝑗∗ , 𝑣2𝑗∗ ,𝑣3𝑗∗ ,𝑢1𝑗∗ ,𝑢2𝑗∗ ) = 𝟎5 for all 𝑝𝑗 ∉ 𝐶𝑃𝑛𝑒𝑤. The CG approach results in 𝑣1∗ = 𝑣3∗ = 0 which shows that two criteria, software and support costs, are dismissed from the compuations because their coresponding weights are zero. In other words, if one drops 𝑥1 and 𝑥3 from the dataset and solve model (4), then the same composite project will be selected. In our approach, the optimal input-output weights for some selected proposals are zero, neverthelsess, none of the factors are completely eleminated from the computations. 22

Table 2. Optimal weights in different approaches The CG approach Virtual DMU

𝑣1∗

𝑣2∗

𝑣3∗

𝑢1∗

𝑢2∗

0.00000

0.00073

0.00000

0.00042

0.01596

The New approach Selected proposals 𝒑𝟏

𝑣1𝑗∗

𝑣2𝑗∗

𝑣3𝑗∗

𝑢1𝑗∗

𝑢2𝑗∗

0.00000

0.00000

0.00847

0.00000

0.13559

𝒑𝟒

0.00000

0.00000

0.01220

0.00000

0.19512

𝒑𝟓

0.00003

0.00000

0.01207

0.05263

0.00000

𝒑𝟕

0.00001

0.00782

0.00071

0.00000

0.20000

𝒑𝟗

0.00000

0.00562

0.00000

0.00000

0.12500

𝒑𝟏𝟎

0.00048

0.00000

0.00000

0.05197

0.00313

𝒑𝟏𝟏

0.00029

0.00000

0.00001

0.03226

0.00000

𝒑𝟏𝟐

0.00000

0.00508

0.00000

0.03111

0.00000

𝒑𝟏𝟓

0.00001

0.00782

0.00071

0.00000

0.20000

𝒑𝟐𝟎

0.00000

0.00000

0.00935

0.00000

0.14953

5. Conclusion Evaluating and selecting a set of proper proposals from a large pool of proposals suggested to an organization is a challenging decision-making problem that has been undertaken in some studies. Addressing this problem using the DEA approach in the presence of resource restrictions was first discussed by Cook & Green (2000). Their approach considers a composite project as a virtual (dummy) unit and then measures its aggregate-efficiency score without dealing with the CCR-efficiency score of individual proposals. In this paper, we develop an alternative DEA approach with the aim of finding a composite project with the highest average CCR-efficiency score of individual involved proposals. Moreover, the CG approach devotes a common set of weights to inputs and 23

outputs of all proposals meanwhile our new approach assigns different weights to those of each individual proposal. A case study of Iran ministry of commerce is taken as an example to illustrate the applicability and efficacy of the suggested approach. The extension of the current methodology to a non-radial measure-based model is an interesting further research avenue. Such a model will exhibit new theoretical challenges, however, it could help resolve issues related to slack variables that are not yet fully enclosed. The developed approaches in the literature for selecting proposals do not take into account the influence of data uncertainties on the evaluation. Hence, devising a robust project selection method is another motivating research direction (see Toloo & Mensah, 2019).

REFERENCES Albino, V., & Gravelli, A. (1998). A neural network application to subcontractor rating in construction firms. International Journal of Project Management, 16(1), 9–14. Asosheh, A., Nalchigar, S., & Jamporazmey, M. (2010). Information technology project evaluation: An integrated data envelopment analysis and balanced scorecard approach. Expert Systems with Applications, 37(8), 5931–5938. Badri, M. A., Davis, D., & Davis, D. (2001). A comprehensive 0–1 goal programming model for project selection. International Journal of Project Management, 19(4), 243–252. Beasley, J. . (2003). Allocating fixed costs and resources via data envelopment analysis. European Journal of Operational Research, 147(1), 198–216. Bhattacharyya, R., Kumar, P., & Kar, S. (2011). Fuzzy R&D portfolio selection of interdependent projects. Computers & Mathematics with Applications, 62(10), 3857– 3870. Chang, P.-T., & Lee, J.-H. (2012). A fuzzy DEA and knapsack formulation integrated model for project selection. Computers & Operations Research, 39(1), 112–125. Charnes, A., Cooper, W. W., & Rhodes, E. (1978). Measuring the efficiency of decision 24

making units. European Journal of Operational Research, 2(6), 429–444. Chen, C.-T., & Cheng, H.-L. (2009). A comprehensive model for selecting information system project under fuzzy environment. International Journal of Project Management, 27(4), 389–399. Cook, W. D., & Green, R. H. (2000). Project prioritization: A resource-constrained data envelopment analysis approach. Socio-Economic Planning Sciences, 34(2), 85–99. Cook, W. D., & Zhu, J. (2007). Classifying inputs and outputs in data envelopment analysis. European Journal of Operational Research, 180(2), 692–699. Deng, H., & Wibowo, S. (2008). Intelligent Decision Support for Evaluating and Selecting Information Systems Projects. Engineering Letters, 16(3), 412–418. Dia, M. (2009). A portfolio selection methodology based on data envelopment analysis. Infor, 47(1), 71–79. Jafarzadeh, H., Akbari, P., & Abedin, B. (2018). A methodology for project portfolio selection under criteria prioritisation, uncertainty and projects interdependency – combination of fuzzy QFD and DEA. Expert Systems with Applications, 110, 237–249. Kapelko, M. (2018). Measuring inefficiency for specific inputs using data envelopment analysis: evidence from construction industry in Spain and Portugal. Central European Journal of Operations Research, 26(1), 43–66. Karasakal, E., & Aker, P. (2017). A multicriteria sorting approach based on data envelopment analysis for R&D project selection problem. Omega (United Kingdom), 73, 79–92. Khoshnevis, P., & Teirlinck, P. (2018). Performance evaluation of R&D active firms. SocioEconomic Planning Sciences, 61, 16–28. Kumar, U. D., Saranga, H., Ramírez-Márquez, J. E., & Nowicki, D. (2007). Six sigma project selection using data envelopment analysis. TQM Magazine, 19(5), 419–441. Lim, S., Oh, K. W., & Zhu, J. (2014). Use of DEA cross-efficiency evaluation in portfolio 25

selection: An application to Korean stock market. European Journal of Operational Research, 236(1), 361–368. Lorie, J. H., & Savage, L. J. (1955). Three problems in rationing capital. The Journal of Business, 28(4), 229–239. Lozano, S. (2013). Using DEA to find the best partner for a horizontal cooperation. Computers and Industrial Engineering, 66(2), 286–292. Martinovic, N., & Savic, G. (2019). Staff assignment to multiple projects based on DEA efficiency. Engineering Economics, 30(2), 163–172. Park, H., Lee, J. (Jay), & Kim, B.-C. B. . (2015). Project selection in NIH: A natural experiment from ARRA. Research Policy, 44(6), 1145–1159. Pérez, F., Gómez, T., Caballero, R., & Liern, V. (2018). Project portfolio selection and planning with fuzzy constraints. Technological Forecasting and Social Change, 131, 117–129. Shakhsi-Niaei, M., Torabi, S. A., & Iranmanesh, S. H. (2011). A comprehensive framework for project selection problem under uncertainty and real-world constraints. Computers and Industrial Engineering, 61(1), 226–237. Sowlati, T., Paradi, J. C., & Suld, C. (2005). Information systems project prioritization using data envelopment analysis. Mathematical and Computer Modelling, 41(11), 1279–1298. Tavana, M., Keramatpour, M., Santos-Arteaga, F. J., & Ghorbaniane, E. (2015). A fuzzy hybrid project portfolio selection method using Data Envelopment Analysis, TOPSIS and Integer Programming. Expert Systems with Applications, 42(22), 8432–8444. Tavana, M., Khalili-Damghani, K., & Sadi-Nezhad, S. (2013). A fuzzy group data envelopment analysis model for high-technology project selection: A case study at NASA. Computers and Industrial Engineering, 66(1), 10–23. Toloo, M., & Mensah, E. K. (2019). Robust optimization with nonnegative decision 26

variables: A DEA approach. Computers & Industrial Engineering, pp. 313–325. Toloo, M., Nalchigar, S., & Sohrabi, B. (2018). Selecting most efficient information system projects in presence of user subjective opinions: a DEA approach. Central European Journal of Operations Research, 26(4), 1027–1051. Vitner, G., Rozenes, S., & Spraggett, S. (2006). Using data envelope analysis to compare project efficiency in a multi-project environment. International Journal of Project Management, 24(4), 323–329. Weingartner, H. M. (1966). Capital Budgeting of Interrelated Projects : Survey and Synthesis. Management Science, 12(7), 485–516. Wu, J., Zhu, Q., Cook, W. D., & Zhu, J. (2016). Best cooperative partner selection and input resource reallocation using DEA. Journal of the Operational Research Society, 67(9), 1221–1237. Yang, C.-L., Chiang, S.-J., Huang, R.-H., & Lin, Y.-A. (2013). Hybrid decision model for information project selection. Quality & Quantity, 47(4), 2129–2142.

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