The Relationship Of Multiple Objectives Linear Programming and Data ...

Australian Journal of Basic and Applied Sciences, 5(11): 1711-1714, 2011 ISSN 1991-8178 

The Relationship Of Multiple Objectives Linear Programming and Data Envelopment Analysis Nasruddin Hassan, Maryam Mohaghegh Tabar School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia 43600 UKM Bangi Selangor D.E., Malaysia. Abstract: The concept of efficiency as it applies to Decision Making Units (DMUs), its solutions and alternatives, plays an important role both in Data Envelopment Analysis (DEA) and Multiple Objective Linear Programming (MOLP). Despite this common concept and other apparent similarities, DEA and MOLP research has developed separately. In this paper the relationships between MOLP and DEA is discussed. Key words: Data envelopment analysis, efficiency score, multiple objective linear programming. INTRODUCTION Data Envelopment Analysis (DEA) was originally proposed by Charnes et al., (1978) as a method for evaluating the relative efficiency of Decision Making Units (DMUs) performing essentially the same task. Each of the units uses multiple inputs to produce multiple outputs. Since the original publication, DEA has become a popular method for analyzing the efficiency of various organizational units. Interestingly, Charnes and Cooper have also had a significant impact on the development of Multiple Objective Linear Programming (MOLP) through the development of Goal Programming (Charnes and Cooper (1961). Since the 1970s, MOLP has become a popular approach for modeling and analyzing certain types of multiple criteria decision problems. Although Charnes and Cooper have played a significant role in the development of DEA and MOLP, researchers in these two camps have generally not paid much attention to research performed in the other camp. Neither have Charnes nor Cooper attempted to tie the two fields together. This is unfortunate, because-despite differences in terminology-DEA and MOLP address similar problems and are structurally very close to each other. In both models, technically speaking, the purpose is to identify efficient points in a certain space and suggest projections of inefficient points on the basis of such information. Literature Survey: In DEA, the projection is performed by letting some mathematical program determine weights that (in the primal case) associate the analyzed point with the best possible efficiency score. In MOLP, the direction of the projection is based on the use of weights (more generally, parameters), which the decision maker (DM) can directly or indirectly influence reflecting his/her preference structure (Nasruddin and Sri, 2009). Furthermore, generally MOLP, and other Multiple Criteria Decision Making (MCDM) methods are considered to be ex ante planning tools, whereas DEA is considered to be an ex post evaluation tool. Some researchers have pointed out that DEA itself is a multiple criteria decision analysis (MCDA) method, where multiple inputs and outputs function as multiple criteria (Jahanshahloo and Foroughi, 2005; Korhonen and Syrjanen, 2004; Li and Reeves, 1999). Others have noted common elements in DEA and multiple criteria analysis methods (Nasruddin et al., 2010a) and have combined the two approaches in identifying the most efficient firms (Belton, 1992; Belton and Stewart, 1999; Belton and Vickers, 1993; Joro et al., 1998; Nasruddin et al., 2010b). Review of Data Envelopment Analysis: Suppose we have n DMUs {DMU j , j  1,2,..., n} , which produce s outputs utilizing m

inputs

y rj : r  1, 2, …, s by

xij : i  1, 2, …, m . Relative efficiency is defined as the ratio of total weighted outputs to

the total weighted inputs. The efficiency measure for

DMUo is defined as

s

o 

u r 1 m

r

y ro

v x i 1

i

io

Corresponding Author: Nasruddin Hassan, School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia 43600 UKM Bangi Selangor D.E., Malaysia.

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Aust. J. Basic & Appl. Sci., 5(11): 1711-1714, 2011 

where the weights and are non-negative. To estimate the DEA efficiency of

DMUo

, we use the following

original DEA model of Charnes et al. (1978) :

where

  0 is a non-Archimedean constant.

Review of Multi Objective Linear Programming: In mathematical terms, the multiobjective optimization problem which is the general form of MOLP can be written as (Yun et al., 2001):

min x

f ( x)  ( f 1 ( x),..., f m ( x)) T

s.t. x  S  {x   n g j ( x)  0, j  1,..., l}, where x  ( x1 ,..., x n ) ) is a design variable and S is the set of all feasible solutions. In general, unlike T

traditional optimization problems with a single objective function, an optimal solution in the sense that minimizes all objective functions f i ( x)(i  1,..., m) simultaneously does not necessarily exist. Hence, the concept of an optimal solution based on the relation of Pareto domination is defined as follows by Yun et al., ( 2001): Definition (Pareto Optimal Solution): A point xˆ  S is said to be a Pareto optimal solution to the multiobjective optimization problem if there exists no x  S such that f ( x )  f ( xˆ ) . The Relationship Between DEA and MOLP: Joro et al., (1998) showed that structurally the DEA formulation to identify efficient units is quite similar to the MOLP model based on the reference point or the reference direction approach to generate efficient solutions. DEA and MOLP should not be seen as substitutes, but rather as complements. They showed that MOLP provides interesting extensions to DEA and DEA provides new areas of application to MOLP. Yun et al., (2001) proposed a method combining generalized data envelopment analysis (GDEA) and GA for generating efficient frontiers in multiobjective optimization problems. His proposed method can yield desirable efficient frontiers even in non-convex problems as well as convex problems. Jahanshahloo and Foroughi (2005) started with analyzing efficiency of an alternative in a multi objective problem. By combination of concepts in data envelopment analysis (DEA) and multiple objective linear programming (MOLP), they searched a composite alternative to compare with a given alternative and assess its efficiency in an MOLP problem. In addition to some results, this leads us to produce some procedures for generating an initial efficient extreme point to an MOLP problem. They considered some models of DEA as MOLP problems. Neto and Meza (2007) compared three families of data envelopment analysis (DEA) models: the traditional radial, the preference structure and the multi-objective models. Lotfi et al.(2010) established an equivalence model between DEA and MOLP and showed how a DEA problem can be solved interactively by transforming it into MOLP formulation. They used Zionts–Wallenius (Z–W) method to reflect the DM’s preferences in the

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process of assessing efficiency. A case study was carried out to illustrate how DEA-oriented efficiency analysis can be conducted using the MOLP method. Review of Methodologies: In this section we compare some methods of relationship between DEA and multiobjective optimization which is the general form of MOLP. Some researchers have pointed out that DEA itself is a multiobjective optimization and others have noted common elements in DEA and multiobjective optimization and have combined the two approaches in identifying the most efficient firm. In the first category, consider the method proposed by Joro et al., (1998). They compared the CCR–O model and that of the reference point model and showed reference point model as a CCR-O model and vice versa. Reference Point Model

CCR-O Model

max    (1 s  1 s )

max    1T s 

s.t.

s.t.



T

T



Y -  y o  s  0

Y -  W  s   g

X

X



 s   xo

 s  b

, s  , s   0  0

, s  , s   0  0

Yun et al., (2001) proposed the GDEA model to solve the multiobjective optimization problem:

max  s.t. m ~   d j    vi ( Fi ( x o )  Fi ( x j )),

j  1,..., p,

i 1

m

v i 1

 1, vi   ,

i

i  1,..., m,

~

where d j  max i 1,...,m {vi (  Fi ( x )  Fi ( x ))}.  is a sufficiently small number and o

j

 is the value of

a monotonically decreasing function with respect to the number of generation. The proposed method is summarized as: Step 1 (Initialization). Generate p-individuals randomly. Here, the number of p is given prior. Step 2 (Crossover _ Mutation). Make p=2-pairs randomly among the population. Making crossover each pair generates a new population. Mutate them according to the given probability of mutation. Step 3 (Evaluation of fitness by GDEA). Evaluate the GDEA-efficiency by solving the problem. Step 4 (Selection). Select p-individuals from current population on the basis of the fitness given by the GDEA-efficiency. The process Step 2 to Step 4 is continued until the number of generations attains a given number. The second category is related to the problems that DEA problem can be solved by transferring it into multiobjective optimization problem. Lotfi et al., (2010) established the equivalence relationship between the output-oriented CCR dual model and the min-ordering formulation as a MOLP formulation as follows: Min-ordering Model

Output –oriented CCR Model max  o

max    f r ( ) r  1,.., s

s.t. n

 j 1

j

y rj   o y ro

 

r  1, 2 ,..., s ,

n

   o  { :   j x ij  x io , i  1,..., m ;  j  0 , j  1,..., n} j 1

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After establishing the equivalency, Lotfi et al., (2010) translated the CCR model to the MOLP and used the method of Zionts-Wallenius to solve it. Conclusion: In this paper different methods have been discussed for evaluation the relationship between DEA and MOLP. Some researchers have pointed out that DEA itself is a multiobjective optimization and others have noted common elements in DEA and MOLP and have combined the two approaches in identifying the most efficient firm. ACKNOWLEDGEMENT We are indebted to Universiti Kebangsaan Malaysia for funding this research under the grant UKM-GUP2011-159. REFERENCES Belton, V. and S.P. Vickers, 1993. Demystifying DEA – a visual interactive approach based on multiple criteria analysis. Journal of the Operational Research Society, 44(9): 883-896. Belton, V. and T.J. Stewart, 1999. DEA and MCDA: Competing or complementary. In: Meskens, N., Roubens, M. (Eds.). Advances in Decision Analysis, 87-104. Belton, V., 1992. Integrating data envelopment analysis with multiple criteria decision analysis. In: Goicoechea, A., Duckstein, L., Zionts, S. (Eds.). Proceedings of the 9th International Conference on Multiple Criteria Decision Making, 71–79. Charnes, A. and Cooper W.W. 1961. Management Models and Industrial Applications of Linear Programming. New York: John Wiley. Charnes, A., Cooper W.W. and R. Rhodes 1978. Measuring the efficiency of decision making units, European Journal of Operational Research, 2(6): 429-444. Jahanshahloo, G.R. and A.A. Foroughi, 2005. Efficiency analysis, generating an efficient extreme point for an MOLP and some comparisons. Applied Mathematics and Computation, 162: 991-1005. Joro, T., P. Korhonen and J. Wallenius, 1998. Structural Comparison of Data Envelopment Analysis and Multiple Objective Linear Programming, Management Science, 44(7): 962-970. Korhonen, P. and M. Syrjanen, 2004. Resource allocation based on efficiency analysis. Management Science 50 (8): 1134-1144. Li, X.B. and G.R. Reeves, 1999. A multiple criteria approach to data envelopment analysis. European Journal of Operational Research, 115(3): 507-517. Lotfi, F.H., G.R. Jahanshahloo, M. Soltanifar, A. Ebrahimnejad and S.M. Mansourzadeh, 2010. Relationship between MOLP and DEA based on output-orientated CCR dual model. Expert Systems with Applications 37(6): 4331-4336. Nasruddin, H. and B.M.B. Sri, 2009. A Goal Programming Model for Scheduling Political Campaign: A Case Study in Kabupaten Kampar, Riau, Indonesia. Journal of Quality Measurement and Analysis (JQMA), 5(2):99-107. Nasruddin, H., M.T. Maryam and S. Parvaneh, 2010a. Resolving multi objectives resource allocation problem based on inputs and outputs using data envelopment analysis method. Australian Journal of Basic and Applied Sciences(AJBAS). ISSN 1991-8178. 4(10): 5320-5325. Nasruddin, H., M.T. Maryam and S. Parvaneh, 2010b. A ranking model of data envelopment analysis as a centralized multi objective resource allocation problem tool. Australian Journal of Basic and Applied Sciences(AJBAS). ISSN 1991-8178. 4(10): 5306-5313. Neto, J.Q.F. and L.A. Meza, 2007. Alternative targets for data envelopment analysis through multiobjective linear programming: Rio de Janeiro Odontological Public Health System Case Study. Journal of the Operational Research Society 58: 865-873. Yun, Y.B., H. Nakayama, T. Tanino and M. Arakawa, 2001. Generation of efficient frontiers in multiobjective optimization problems by generalized data envelopment analysis. European Journal of Operational Research 129: 586-595.

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